Title: Math and science / Analysis / Tao / 09. Continuous functions on R

Title: Intervals (of the reals), definition
Let \( a, b \in \mathbb{R}^* \) be extended real numbers.
We define the closed interval \( [a, b] \) by
\[ [a, b] := \{ x \in \mathbb{R}^* : a \leq x \leq b\} \]
We define the half-open intervals \( [a, b) \) and \( (a, b] \) by
\( [a, b) := \{x \in \mathbb{R}^* : a \leq x \leq b\}; \)  and \( (a, b] := \{x \in \mathbb{R}^* : a \le x \leq b \} \) 
And we define the open interval \( (a, b) \) by
\[ (a, b) := \{ x \in \mathbb{R}^* : a < x < b \} \]
We call \( a \) the left endpoint and \( b \) the right endpoint.
BackOnlyText: If \( a \) and \( b \) are real numbers (not \( \infty \) or \( -\infty \)), then all intervals above are subsets of the real line.
The real line itself is the open interval \( (-\infty, \infty) \), and the extended real line is the closed interval \( [-\infty, \infty] \).
We sometimes refer to intervals where one endpoint is infinite as being a half-infinite interval, and intervals where both endpoints are infinite as being a double-infinite interval; all other intervals are bounded intervals.
Source: p213

Title: Closure, definition
To define a closure, we will utilize ε-adherent points and adherent points. Sets of reals have adherent points analogous to sequences of reals having limit points.
ε-adherent point
Let \( X \) be a subset of \( \mathbb{R} \), let \( \varepsilon > 0 \) be a real and \( x \in \mathbb{R} \) be another real. We say that \( x \) is ε-adherent to \( X \) iff there exists a \( y \in X \) which is ε-close to \( x \) (i.e. \( |x - y| \leq \varepsilon \) ).
Adherent point
Let \( X \) be a subset of \( \mathbb{R} \), and let \( x \in \mathbb{R} \) be a real. We say that \( x \) is an adherent point of \( X \) iff it is ε-adherent to \( X \) for every \( \varepsilon > 0 \).
Closure
Let \( X \) be a subset of \( \mathbb{R} \). The closure of \( X \), sometimes denoted as \( \overline{X} \), is defined to be the set of all adherent points of \( X \).
Example: The number 1 is ε-adherent to the open interval (0, 1) for every \( \varepsilon > 0 \), and is thus an adherent point of the interval (0, 1). The number 2, in comparison is not 0.5-adherent to (0,1), so can't be an adherent point of the interval.
The closure of \( \mathbb{N} \) is \( \mathbb{N} \). The closure of \( \mathbb{Z} \) is \( \mathbb{Z} \). The closure of \( \mathbb{Q} \) is \( \mathbb{R} \). And the closure of \( \mathbb{R} \) is \( \mathbb{R} \). The closure of \( \emptyset \) is \( \emptyset \).
Closures of intervals
- Closure of \( (a, b), [a, b), (a, b], \text{ and } [a, b] \) is \( [a, b] \)
- Closure of \( (a, \infty) \) or \( [a, \infty) \) is \( [a, \infty) \)
- Closure of \( (-\infty, a) \) or \( (-\infty, a] \) is \( (-\infty, a] \)
- Closure of \( (-\infty, \infty) \) is \( (-\infty, \infty) \)
Source: p214

Title: Closed set (of reals), definition
Q: A subset \( E \subseteq \mathbb{R} \) is said to be closed if [...].
A: \( \overline{E} = E \)
BackOnlyText: In other words, \( E \) contains all of its adherent points. The operation which is implicit is the limit; any limt constructed from elements of \( E \) will equal an element in \( E \).
Example: \( [a, b], (-\infty, b] \), \( [a, \infty) \text{ and } (-\infty, \infty) \) are closed, while \( (a, b), [a, b), (a, b], (-\infty, b), (a, \infty ) \) are not.
\( \mathbb{N}, \mathbb{Z}, \mathbb{R} \text{ and } \emptyset \) are closed, while \( \mathbb{Q} \) is not.
Source: p216

Title: Limit points and isolated points (of sets of reals), definition
Q: Limit points
Let \( X \) be a subset of the real line. We say that \( x \) is a limit point (or cluster point) of \( X \) if [...].
Not to be confused with limit points of sequences.
Isolated points
Let \( X \) be a subset of the real line. We say that \( x \) is an isolated point of \( X \) if \( x \in X \) and [...].
A: it is an adherent point of \( X\setminus \{x\} \); there exists some real \( \varepsilon > 0 \) such that \( |x - y| > \varepsilon \) for all \( y \in X\setminus \{x\} \)
Example: 3 is an adherent point of the set \( X = (1, 2) \cup \{3\} \), but it is not a limit point of \( X \), since it is not adherent to \( X \setminus \{3\} \). Instead, 3 is an isolated point of \( X \). In comparison, 2 is a limit point, but not an isolated point of \( X \).
Source: p216

Title: Bounded sets (of reals)
Q: A subset \( X \) of the real line is said to be bounded if [...].
A: we have \( X \subseteq [-M, M] \) for some real number \( M > 0 \)
BackOnlyText: Not to be confused with bounded sequences.
Example: The interval \( [a, b] \) is bounded for any real numbers \( a \) and \( b \), as it is contained inside \( [-M, M] \) where \( M := max(|a|, |b|) \).
The half-infinite intervals and the doubly infinite interval are not bounded.
The sets \( \mathbb{N}, \mathbb{Z}, \mathbb{Q} \text{ and } \mathbb{R} \) are all unbounded.
Source: p217

Title: Heine-Borel theorem for the line (Tao)
Q: Let \( X \) be a subset of \( \mathbb{R} \). Then the following two statements are equivalent:
1. [...].
2. Given any sequence \( (a_n)_{n=0}^{\infty} \) of real numbers which takes values in \( X \) (i.e., \( a_n \in X \text{ for all } n\) ), there exists a [...].
Tao introduces Heine-Borel theorem quite separate to the Bolzano-Weierstrass theorem. I think that the Heine-Borel theory is more fitting to be grouped with the Bolzano-Weierstrass theorem; it is a theorem of sequences and not directly concerned with continuous functions (chapter 9, where it appears).
A: \( X \) is closed and bounded; subsequences \( (a_{n_j})_{j=0}^{\infty} \) of the original squence, which converges to some number \( L \) in \( X \)
BackOnlyText: It's worth comparing this to the Bolzano-Weierstrass theorem. The Bolzano-Weierstrass theoem is (almost) enough to show the implication from a) to b) above. The main addition of the Hiene-Borel theorem is in the implication from b) to a). In fact, it requires the axiom of choice. (Well, Tao's proof uses it).
In the language of metric space topology, this is asserting that every subset of the real line which is closed and bounded is also compact.
There exists a more general version of this theorem, due to Eduard Heine (1821-1881) and Emile Borel (1871-1956).
Source: p217

Title: Arithmetic operations on functions
Q: Given two functions \( f : X \to \mathbb{R} \) and [...], we can define their:
sum, \( f + g : X \to \mathbb{R} \)
\( (f + g)(x) := f(x) + g(x) \)
difference, \( f - g : X \to \mathbb{R} \)
\( (f - g)(x) := f(x) - g(x) \)
maximum, \( \max(f, g) : X \to \mathbb{R} \)
\( \max(f, g)(x) := \max(f(x), g(x)) \)
minimum, \( \min(f, g) : X \to \mathbb{R} \)
\( \min(f, g)(x) := \min(f(x), g(x)) \)
product, \( fg : X \to \mathbb{R} \) (or \( f \cdot g \) )
\( (fg)(x) := f(x)g(x) \)
quotient, assuming \( g(x) \ne 0 \text{ for all } x \in X\), \( f/g : X \to \mathbb{R} \)
\( (f/g)(x) := f(x)/g(x) \)
constant multiple, \( cf : X \to \mathbb{R} \)
\( (cf)(x) := cf(x) \)
A: \( g : X \to \mathbb{R} \) (from the same set \( X \) to the reals)
BackOnlyText: There is really nothing special going on here. Not even any composition. All of these could apply to functions from some set to the rationals also. In general, when two functions share a domain and range, for every operation present in the range it should be possible to define a new function where each element of the domain is mapped to the value that is the result of the operation applied to the separate function outputs. Limitations may need to be applied, like in the case of the quotient.
I considered not adding this card, but I supposed that it would be good to become familiar with the symbols and comfortable with how rudimantary they are.
Source: p220

Title: Function convergence at a point, definition
ε-closeness, of a function to a real
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( L \) be a real, and let \( \varepsilon > 0 \) be a real.
We say that the function \( f \) is ε-close to \( L \) iff \( f(x) \) is ε-close to \( L \) for every \( x \in X \).
Local ε-closeness, of a function to a real
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( x_0 \) be an adherent point of \( X \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real.
We say that \( f \) is ε-close to \( L \) near \( x_0 \) iff there exists a \( \delta > 0 \) such that \( f \) becomes ε-close to \( L \) when restricted to the set \( \{x \in X : |x - x_0| < \delta \} \).
Convergence of a function at a point
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real.
We say that \( f \) converges to \( L \) at \( x_0 \) in \( E \), and write \( \lim_{x \rightarrow x_0; x \in E} f(x) = L\), iff \( f \), after restricting to \( E \), is ε-close to \( L \) near \( x_0 \) for every \( \varepsilon > 0 \). If \( f \) does not converge to any number \( L \) at \( x_0 \), we say that \( f \) diverges at \( x_0 \), and leave \( \lim_{x \rightarrow x_0; x \in E} f(x) \) undefined.
These definitions are separated into 2 paragraphs due to the many supporting objects required.
BackOnlyText: Laxness of the phrase "restricting the function \( f \) to the range \( X \)"
Tao talks on page 219 about the laxness of this phrase. What is mean is that we are referring to another distinct function \( f|_X : X \to B \) defined by \( f|_X(x) = f(x) \).
The familiar structure of closeness
We are getting used to the proceedure now: define a notion of ε-closeness, a notion of restricted ε-closeness, and finally, a notion of having restricted ε-closeness for all ε.
Correction to image: either \( x_0 \notin X \) or \( f(x_0) = Q \). If \( f(x_0) \) is something other than \( Q \), then \( f \) wouldn't be locally close to \( Q \) near \( x_0 \) for any \( \varepsilon > |Q - f(x_0)| \).
Example: Let \( f : [1, 3] \to \mathbb{R} \) be the function \( f(x) = x^2 \).
- This function is 5-close to 4.
- It is not 0.1 close to 4.
- it is 0.1 close to 4 near 2 (when \( f \) is restricted to the set \( \{ x \in [1, 3] : |x - 2| < 0.01 \} \) it is 0.1 close to 4).
- It is 0.1 close to 9 near 3. (Notice the half-open interval being used.)
- It converges to 9 at 3 in \( [1,3] \) (when not restricted at all).
Source: p221-222

Title: Function convergence's equivalence to sequence convergence
Q: Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real. Then the following two statements are logically equivalent:
- \( f \) converges to \( L \) at \( x_0 \) in \( E \).
- [...].
A: For every sequence \( (a_n)_{n=0}^{\infty} \) which consists entirely of elements of \( E \) and converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( L \)
BackOnlyText: In view of the above proposition, we will sometimes write "\( f(x) \to L\) as \( x \to x_0 \) in \( E \)" or "\( f \) has a limit \( L \) at \( x_0 \) in E" instead of "\( f \) converges to \( L \) at \( x_0 \) in \( E \)" or \( \lim_{x\rightarrow x_0;x\in E}^{\infty} f(a_n) = L \).
Use limits of sequences to calculate limits of functions
This proposition is very important in allowing us to determine what real a function converges to without having to manipulate the definition of function convergence and play around with ε-closeness. Instead we can use all of our built up knowledge of limits of sequences.
Source: p223

Title: Limit laws for functions, proposition
Q: Combining the rudimentary definitions of arithmetic operations on functions with the weildy definition of function limits gives us the limit laws of functions.
As words, they are as follows:
Let \( X \) be a subset of \( \mathbb{R} \), let \( E \) be a subset of \( X \), let \( x_0 \) be [...] of \( E \) and let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions. Suppose that \( f \) has a limit \( L \) at \( x_0 \) in \( E \), and \( g \) has a limit \( M \) at [...] in \( E \), then:
- \( f+g \) has a limit \( L + M \) at \( x_0 \) in \( E \)
- \( f-g \) has a limit \( L - M \) at \( x_0 \) in \( E \)
- \( \max(f, g) \) has a limit \( \max(L, M) \) at \( x_0 \) in \( E \)
- \( \min(f, g) \) has a limit \( \min(L, M) \) at \( x_0 \) in \( E \)
- \( fg \) has a limit \( LM \) at \( x_0 \) in \( E \)
- \( cg \) has a limit \( cL \) at \( x_0 \) in \( E \) (assuming if \( c \) is a real number)
- \( f/g \) has a limit \( L / M \) at \( x_0 \) in \( E \) (assuming \( g(x) \ne 0 \text{ for all } x \in E \) and \( M \ne 0 \))
A: an adherent point; \( x_0 \)
BackOnlyText: Each of these can be expressed more informally (losing reference to sets \( X \), \( \mathbb{R} \) and \( E \) and dropping the specification of restrictions) like so (just one example here):
\[ lim_{x \rightarrow x_0}(f/g)(x) = \frac{\lim_{x \rightarrow x_0} f(x)}{\lim_{x \rightarrow x_0} g(x)} \]
Where does the effort lie in these propositions
I find it easier to think about the above equation going from right to left, in the same direction as the worded version above proceed. One way of thinking about these propsotions is that when both \( f \) and \( g \) satisfy the criteria to have a limit at \( x_0 \) in \( E \), then so too does the function \( (f+g) \). The value of the limit of \( (f+g) \) at \( x_0 \) in \( E \) must be the sum of the limits of \( f \) and \( g \). The latter idea follows intuatively once the former idea (that \( (f+g) \) must have a limit) is accepted.
Source: p225

Title: Calculating function convergence
Q: 1. A symbolic expression
Consider the very simple, somewhat informal, expression:
\[ \lim_{x \rightarrow x_0; x\in \mathbb{R}} c \]
2. The exact expression
Expressed exactly according to the wording of the definition of function convergence, the above expression represents the object:
The real to which the function \( f: \mathbb{R} \to \mathbb{R} \), defined by \( f(x) := c \), [...].
I would argue that symbolic form is only exact when \( c \) is replaced with the explicit function:
\[ \lim_{x \rightarrow x_0; x \in \mathbb{R}} f(x) \]
The definition is paired with a proof that such a number is unique, so the use of 'the' instead of 'a' is justified.
3. Unwrapping the definition of function convergence.
If we expand the meaning of "converges to <\( i \)>, at <\( x_0 \)> on <\( E \)>" according to the definition of convergence, the object becomes:
The real to which the function \( f: \mathbb{R} \to \mathbb{R} \), defined by \( f(x) := c \), when restricted to \( \mathbb{R} \), is ε-close to [...] for [...].
"when" was introduced into the wording, as without it it was not clear that "is ε-close to near \( x_0 \)" means "is ε-close to <the object> near \( x_0 \)".
4. We can check, but not compute
The statement in 3 allows us to check if a certain real \( L \) matches the criteria to be able to be equated with the expression in 1. This isn't that useful, as you kind of need to know what the value is ahead of time. For example, we can check that the real \( c \) matches the criteria:
For any \( \varepsilon > 0 \), we can find a \( \delta > 0 \) such that when \( f \), restricted to \( \mathbb{R} \), is further restricted to \( \{x \in \mathbb{R} : |x - x_0| < \delta \} \), the function is ε-close to \( c \). For example, let \( \varepsilon > 0 \) be arbitary. Choose \( \delta = 1 \), and consider \( f \) restricted to \( \{x \in \mathbb{R} : |x - x_0| < 1 \} = (x_0 - 1, x_0 + 1) \). For any \( x \) in this set we have \( |f(x) - c| = |c - c| = 0 < \varepsilon \). Thus, \( c \) meets the criteria to be able to equate it to the expression from 1.
5. Using Proposition 9.3.9
What an ordeal... This is why Prop. 9.3.9 is so important. Repeated here it is:
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real. Then the following two statements are logically equivalent:
- \( f \) converges to \( L \) at \( a_0 \) in \( E \).
- For every sequence \( (a_n)_{n=0}^{\infty} \) which consists entirely of elements of \( E \) and converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( L \).
Using this proposition, we can calculate the value of the expression in 1. This is because we are able to express everything in terms of limits of sequences, for which we have already developed a wealth of propositions covering equality, such as Theorem 6.1.19. Using these theorems, we don't need to rederive results like I did above for the constant function \( f(x) := c \).
6. Utilize sequence limit laws
Taking our expression from 1, Prop. 9.3.9 allows us to replace it with the expression:
The object to which any sequence \( (f(a_n))_{n=0}^{\infty} \) converges to as long as the sequence \( (a_n)_{n=0}^{\infty} \) converges to \( x_0 \).
\( (f(a_n))_{n=0}^{\infty} = (c)_{n=0}^{\infty} \), by definition of \( f \). Finally, \( (c)_{n=0}^{\infty} = c \), by Theorem 6.1.19. So:
\[ \lim_{x \rightarrow x_0; x\in \mathbb{R}} c  = c \]
And we can ignore the requirements on sequence \( (a_n)_{n=0}^{\infty} \), as \( f(a_n) \) is independent of \( a_n \)
A: converges to at \( x_0 \) in \( \mathbb{R} \); when near \( x_0 \); every \( \varepsilon > 0 \)
Source: p225

Title: Continous functions
Q: Let \( X \) be a subset of \( \mathbb{R} \), and let \( f: X \to \mathbb{R} \) be a function. Let \( x_0 \) be an element of \( X \). We say that \( f \) is continuous at \( x_0 \) iff we have:
[\[\text{some limit = what?}\]]
in other words, the limit of \( f(x) \) as \( x \) converges to \( x_0 \) in X [...].
We say that \( f \) is continuous on \( X \) (or simply continuous) iff \( f \) is continuous at \( x_0 \) for every \( x_0 \in X \). We say that \( f \) is discontinous at \( x_0 \) iff it is not continous at \( x_0 \).
Tao describes this definition as one of the most fundamental notions in the theory of functions.
Question: what is Dirichlet's function and what is Thomae's function? How do they relate to function continuity?
A: \[ \lim_{x \to x_0; x \in X} f(x) = f(x_0); \]; exists and is equal to \( f(x_0) \)
BackOnlyText: Essence of continuity
A collection of ideas that suggest ways of thinking about function limits and continuity.
Maybe some or all of these would be better on the card for function limits.
Nearness, maintained through function application
Abbott:
Given a function \( f \) with domain \( A \subset \mathbb{R} \), we want to define continuity at a point \( c \in A \) to mean that if \( x \in A \) is chosen near \( c \), then \( f(x) \) will be near \( f(c) \).
Dirichlet's function
To know what something is, it can be best to investigate what it isn't. This line of thinking brings us to some cases that test the notion of continuity.
Peter Lejeune Dirichlet defined the function \( g \) based on whether the input variable is rational or irrational:
\[ g(x) = \begin{cases} 1 &\quad \text{if } x \in \mathbb{Q} \\ 0 &\quad \text{if } x \notin \mathbb{Q} \\ \end{cases} \]
Of this function, does it make sense to give meaning to the following limit: \( \lim_{x \to \frac{1}{2}} g(x) \)? One idea is to consider this to represent the limit of the sequence \( (g(x_n))_{n=0}^{\infty} \). But for this expression, we must decide the sequence \( (x_n)_{n=0}^{\infty} \), and it can be seen that this choice leads different possible sequences \( (g(x_n))_{n=0}^{\infty} \). For example, if each \( x_n \) is rational, then:
\[ \lim_{n\to\infty} g(x_n) = 1 \]
whereas, if each \( x_n \) is irrational, then:
\[ \lim_{n\to\infty} g(x_n) = 0 \]
Our naive definition of function limits is clearly insufficient; if we were to assign a value to the expression \( \lim_{n\to c} g(x) \) in terms of sequences in the domain, we would want the value to be independent of how we approach \( c \). Whatever the definition of the function limit, it seems fair to say that the example above, \( \lim_{x \to \frac{1}{2}} g(x) \), should not be given a value.
Not only at \( \frac{1}{2} \), but at all points of the domain of Dirichlet's we see that the function fails to meet our intuitive idea of continuity. Realize that both \( \mathbb{Q} \) and \( \mathbb{I} \) are both dense in \( \mathbb{R} \), and so any point in \( \mathbb{R} \) can be expressed as a limit of a sequence confined to \( \mathbb{Q} \) or a sequence confined to \( \mathbb{I} \).
Dirichlet's function is said to be a nowhere-continuous function on \( \mathbb{R} \).
Sequence independence
What happens if we change the definition of \( g \) like so:
\[ g(x) = \begin{cases} x &\quad \text{if } x \in \mathbb{Q} \\ 0 &\quad \text{if } x \notin \mathbb{Q} \\ \end{cases} \]
If \( c \) is not zero, then we are in the same situation as before; but if \( c = 0 \), then the sequence \( g(x_n)_{n=0}^{\infty} \) approaches zero in both the case where \( \forall n \in \mathbb{N}, x_n \in \mathbb{Q} \) and where \( \forall n \in \mathbb{N}, x_n \in \mathbb{I} \).
We should be more comfortable considering assigning the value 0 to the expression \( \lim_{x \to 0}g(x) \). This observation gets at the heart of what it should mean for a function limit to exist. We really want
\[ \lim_{x \to c} f(x) = L \quad \text{iff} \quad f(x_n) \to L \; \text{ for any sequences } (x_n) \text{ such that } \lim_{n\to \infty}{x_n} = c \]
Thomae's function
The eponymously named sequence:
\[ t(x) = \begin{cases} 1 &\quad \text{if } x = 0 \\ \frac{1}{n} &\quad \text{if } x = \frac{m}{n} \in \mathbb{Q} \setminus \{0\} \text{ is in lowest terms, with } n > 0 \\ 0 &\quad \text{if } x \notin \mathbb{Q} \\ \end{cases} \]
The last term means that any sequence \( (t(x_n)) \) for irrational sequences \( (x_n) \) will always approach (be) \( 0 \). This includes sequences of irrationals that approach a rational. Thus, the last term insures that \( t \) is not continuous at rational points.
In comparison, the effect of the middle term is to send \( t(x) \) towards zero for sequences of rationals \( (x) \) that approach an irrational. But \( t \) at an irrational point is 0 anyway, so we find that \( t \) seems to be continuous at irrational points.
Thus, Thomae's function has the strange property of being continuous at all irrational points, yet discontinuous at all rational points.
Historical perspective
Abbott mentions in Understanding Analysis that the concept of continuity came long after (~200 years) the derivative was in wide use; the interesting point being that this is now in contrast to the order in which the subjects are taught.
Abbott explains that the concept of a function was, for a long time, more restricted and did not permit discontinuities. When Fourier series (limits of sums of continuous functions) began to be used to represent discontinuous functions, the question arose: does the continuity of the polynomials being summed not imply the continuity of the limit of the sum? Such questions required the idea of continuity to have a definition that was more precise than "no holes".
Example: Constant function
Let \( f : \mathbb{R} \to \mathbb{R} \) be the constant function \( f(x) := c \), for some constant \( c \). Then for every real \( x_0 \in \mathbb{R} \), we have
\[ \lim_{x\to x_0; x\in \mathbb{R}}f(x) = \lim_{x \to x_0; x \in \mathbb{R}}c = c = f(x_0) \]
Thus, \( f \) is continuous at every point \( x_0 \in \mathbb{R} \), or in other words \( f \) is continuous on \( \mathbb{R} \).
Identity function
Let \( f : \mathbb{R} \to \mathbb{R} \) be the identity function \( f(x) := x \), for some constant \( c \). Then for every real \( x_0 \in \mathbb{R} \), we have
\[ \lim_{x\to x_0; x\in \mathbb{R}}f(x) = \lim_{x \to x_0; x \in \mathbb{R}}x = x_0 = f(x_0) \]
Thus, \( f \) is continuous at every point \( x_0 \in \mathbb{R} \), or in other words \( f \) is continuous on \( \mathbb{R} \).
Signum function
Let \( f : \mathbb{R} \to \mathbb{R} \) be the signum function:
\[ sgn(x) := \begin{cases}
1, &\quad \text{if } x > 0 \\
0, &\quad \text{if } x = 0 \\
-1 &\quad \text{if } x < 0 \\
\end{cases} \]
Then \( sgn(x) \) is continuous at every non-zero value of \( x \), for instance , at 1:
\[ \begin{aligned} \\
\lim_{x \to 1; x \in \mathbb{R}} sgn(x) &= \lim_{x \to 1; x \in (0.9, 1.1)} sgn(x) \\
&= \lim_{x \to 1; x \in (0.9, 1.1)} 1 \quad \text{(as limits are local)} \\
&= 1 \\
&= sgn(1) \\
\end{aligned} \]
On the other hand, \( sgn \) is not continuous at 0, since the limit there doesn't exist.
Tao has some other nice examples.
Source: Tao, p228
Abbott, p112

Title: Equivalent formulations of function continuity
Q: Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( x_0 \) be an element of \( X \). Then the following four statements are logically equivalent:
1. \( f \) is continuous at \( x_0 \).
2. [...]
3. [...]
A: For every sequence \( (a_n)_{n=0}^{\infty} \) consisting of elements of \( X \) which converges to \( x_0 \), the sequence \( (f(a_n))_{n=0}^{\infty} \) converges to \( f(x_0) \).; For any real \( \varepsilon > 0 \) there exists a real \( \delta > 0 \) such that \( |f(x) - f(x_0)| < \varepsilon \) for all \(x \in X \) and \( |x - x_0| < \delta \)
BackOnlyText: Pseudo-proof
a) and b) correspond to a) and b) of Prop 9.3.9 (convergence of \( f \) to \( L \) is equivalent to all sequences converging to \( L \)); we just add the idea of continuity to replace \( L \) with \( f(x_0) \).
c) comes about by unwrapping the definition of function convergence giving us local ε-closeness; we just replace \( L \) with \( f(x_0) \).
Importance
Prop 9.3.9 was important in allowing us to determine fuction limits by switching to limits of sequences; this proposition (9.4.7), if the function in question is continuous, allows us to determine its limit, say at \( x_0 \) simply by evaluating \( f(x_0) \). This is far easier than working with the definition of function convergence directly.
In addition, it sets the stage for arithmetic preservation of continuity, which allows us to assert continuity for many functions, which in turn allows us to calculate the limits of functions easily.
[TODO: add a progress->line]
Source: p230

Title: Arithmetic preserves continuity
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions, and let \( x_0 \) be an element of \( X \). Then, if \( f \) and \( g \) are continuous at \( x_0 \), then so too are the functions \( f+g \), \( f-g \), \( \max(f,g) \), \( \min(f,g) \) and \( fg \). If \( g \) is non-zero on \( X \), then \( f/g \) is also continuous at \( x_0 \).
BackOnlyText: Derivation
This proposition comes about by considering the limit laws for functions with the definition of continuity.
Importance
This proposition can be used to prove the continuity of many functions. For example, given the continuity of the identiy function, \( f(x) = x \), at every point in \( \mathbb{R} \), we can show that the function \( g(x) := \frac{max(x^3 + 4x^2 + x + 5, x^4 - x^3)}{(x^2 - 4)} \) is continous at every point in \( \mathbb{R} \) except for two points \( x = +2, x = -2 \).
Below are some more examples. Utilizing the below propositions we can show the continuity of complicated functions like \( h(x):= \frac{|x^2 -8x + 7|^{\sqrt{2}}}{(x^2 + 1)} \).
Exponentiation is continuous, I
Let \( a > 0 \) be a positive real number. Then the function \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x):= a^x \) is continuous.
Exponentiation is continuous, II
Let \( p \) be a real number. Then the function \( f : (0, \infty) \to \mathbb{R} \) given by \( f(x):= x^p \) is continuous.
Absolute value is continuous
The function \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = |x| \) is continuous.
This follows from the continuity of \( max(x, -x) \).
Composition preserves continuity
Let \( X \) and \( Y \) be subsets of \( \mathbb{R} \), and let \( f: X \to Y \) and \( g: Y \to R \) be functions. Let \( x_0 \) be a point in \( X \). If \( f \) is continuous at \( x_0 \) and \( g \) is continuous at \( f(x_0) \), then the composition \( g \circ f : X \to \mathbb{R} \) is continuous at \( x_0 \).
Source: p230-231

Title: Left and right limits
The two separate halves of a complete limit \( \lim_{x \to x_0; x \in X}f(x) \).
Let \( X \) be a subset of \( \mathbb{R} \), \( f: X \to \mathbb{R} \) be a function and \( x_0 \) be a real number.
If \( x_0 \) is an adherent point of \( X \cap (-\infty, x_0) \), then we define the left limit, \( f(x_0-) \) of \( f \) at \( x_0 \) to be:
\[ f(x_0-) := \lim_{x \to x_0; x \in X \cap (-\infty, x_0)} f(x) \]
If \( x_0 \) is an adherent point of \( X \cap (x_0, \infty) \), then we define the right limit, \( f(x_0+) \) of \( f \) to be:
\[ f(x_0+) := \lim_{x \to x_0; x \in X \cap (x_0, \infty)} f(x) \]
Note: if and only if \( x_0 \) is an adherent point of \( (-\infty, x_0) \) then it is a limit point of \( (-\infty, x_0] \), as adherent to a set minus the element in question is the definition of being a limit point of that set.
BackOnlyText: Shorthand notations are:
\[ \lim_{x \to x_0-}f(x) := \lim_{x \to x_0; x \in X \cap (-\infty, x_0)}f(x) \]
\[ \lim_{x \to x_0+}f(x) := \lim_{x \to x_0; x \in X \cap (x_0, \infty)}f(x) \]
Example: Let \( f : \mathbb{R} \to \mathbb{R} \) be the signum function:
\[ sgn(x) := \begin{cases} \\ 1, &\quad \text{if } x > 0 \\ 0, &\quad \text{if } x = 0 \\ -1 &\quad \text{if } x < 0 \\ \end{cases} \]
The \( sgn(x) \) is continuous at every non-zero value of \( x \). The left and right limits at 0 are:
\[ sgn(0-) = \lim_{x \to x_0; x \in X \cap (-\infty, 0)}sgn(x) = \lim_{x \to x_0; x \in X \cap (-\infty, 0)}-1 = -1 \] \[ sgn(0+) = \lim_{x \to x_0; x \in X \cap (0, \infty)}sgn(x) = \lim_{x \to x_0; x \in X \cap (0, \infty)}-1 = 1 \]
Source: p232-233

Title: Left and right limits and discontinuities
Let \( X \subseteq \mathbb{R} \) and let \( x_0 \in X \). Let \( f: X \to \mathbb{R} \) be a function.
If \( f(x_0-) \) and \( f(x_0+) \) both exist and are both equal to \( f(x_0) \), then \( f \) is continuous at \( x_0 \).
\( x_0 \) is an element of \( X \) by definition, so \( f(x_0) \) must exist and be defined as \( f \)'s range includes all of \( X \).
\( x_0 \) must be an adherent point of both \( X \cap (-\infty, x_0) \) and \( X \cap (x_0, \infty) \), as these are requirements for the left and right limits to exist.
Discontinuities
Note that we require both ① limits to exist and ② be equal to each other and ③ to \( f(x_0) \), and ④ that the statement is only an implication, not a bi-implication. Discontinuities can be found by looking at these properties. Four discontinuities that arise at (at least):
1. Asymptotic discontinuity and oscillatory discontinuity
2. Jump discontinuity
3. Removable discontinuity
4. Domain break discontinuity, with function continuity maintained (I made up this name). Not 100% if it exists.
These are discussed further on the back side.
Example: 1. A left or right limit doesn't exist
Two examples of discontinuites that arise from a limit not existing:
Asymptotic discontinuity
One or both limits do not exist as, informally, \( f(x) \) tends to infinity as \( x \to x_0 \). Example: \( f: \mathbb{R} \setminus \{0\}, f(x) := \frac{1}{x} \) has a discontinuity at 0.
Oscillatory discontinuity
A function can remain bounded but still not have a left or right limit near a point. For example: \( f : \mathbb{R} \to \mathbb{R} \) where \( f(x) \) is \( 1 \) if \( x \in \mathbb{Q} \) or \( 0 \) otherwise.
2. Jump discontinuity
Where both limits exist but are not equal, we have a jump discontinuity. At least one must not be equal \( f(x_0) \). For example, \( f: \mathbb{R} \to \mathbb{R} \) defined by:
\[ f(x) := \begin{cases}
0 \quad \text{if } x \le 0 \\
1 \quad \text{if } x > 0 \\
\end{cases} \]
3. Removable discontinuity
Let an right limits can exist and be equal to \( f(x_0) \), yet \( f \) still might not be continuous. For example, \( f: \mathbb{R} \to \mathbb{R} \) defined by:
\[ f(x) := \begin{cases}
1 \quad \text{if } x = 0 \\
0 \quad \text{if } x \neq 0 \\
\end{cases} \]
Then both left and right limits exist and we have \( f(x_0-) = f(x_0+) \neq f(x_0) \). This is called a removable discontinuity.
4. (Not sure what called?) Domain break discontinuity
If \( X = [0, 1] \cap [3, 4] \) and \( f : X \to \mathbb{R} \) is given by \( f(x):= 2x \) (or some other function continuous on \( \mathbb{R} \)), then surprisingly, according to how Tao defines continuity, \( f \) is continuous at all endpoints \( x = 0 \), \( x = 1 \), \( x = 3 \) and \( x = 4 \). This is so as for any \( \varepsilon > 0 \) we can find a \( \delta > 0 \) such that \( |f(i) - f(a)| < \varepsilon \) for all \( i \in \{x \in X \cap (a- \delta, a + \delta)\} \).
Considering this example, continuity might best be thought of as the question of whether for all sequences \( (a_n)_{n=0}^{\infty} \) that converge to \( x_0 \) it can be said that \( (f(a_n))_{n=0}^{\infty} \) converges to \( f(x_0) \). In this view, we are forced to ignore the parts of the domain that are not connected to \( x_0 \) (what is the techical term for not connected?).
Source: p234

Title: The maximum principle
Continuous functions whose domain is a closed enjoy two useful properties:
- the maximum principle
- the intermediate value theorem
This card covers the first. If a function is continuous and it's domain is a closed interval, then it has a maximum value (one or more values of the domain map to a maximum).
Bounded functions
Let \( X \subseteq \mathbb{R} \) and let \( f : X \to \mathbb{R} \) be a function.
We say that \( f \) is bounded from above iff there exists an \( M \in \mathbb{R} \) such that \( f(x) \le M \text{ for all } x \in X \).
We say that \( f \) is bounded from below iff there exists an \( M \in \mathbb{R} \) such that \( f(x) > M \text{ for all } x \in X \).
We say that \( f \) is bounded iff it is both bounded from above and bounded from below.
Continuous functions on a closed interval are bounded
Let \( a < b \) be real numbers. Let \( f : [a,b] \to \mathbb{R} \) be a function continuous on \( [a, b] \). Then \( f \) is a bounded function.
Obtains maximum/minimum
Let \( f: X \to \mathbb{R} \) be a function, and \( x_0 \in X \).
We say that \( f \) obtains its maximum at \( x_0 \) iff \( f(x_0) \ge f(x) \text{ for all } x \in X \).
We say that \( f \) obtains its minimum at \( x_0 \) iff \( f(x_0) \le f(x) \text{ for all } x \in X \).
Maximum principle
Let \( a < b \) be real numbers. Let \( f : [a,b] \to \mathbb{R} \) be a function continuous on \( [a, b] \).
Then there is an \( x_{max} \in [a, b] \) such that \( f(x_{max}) \ge f(x) \text{ for all } x \in [a, b] \) (\( f \) obtains its maximum at \( x_{max} \)).
Similarly, there is an \( x_{min} \in [a, b] \) such that \( f(x_{min}) \le f(x) \text{ for all } x \in [a, b] \) (\( f \) obtains its minimum at \( x_{min} \)).
BackOnlyText: Proof
The proof of the maximum principle is good to study, as it brings together many ideas, including:
- Supremum
- Axiom of choice
- Heine-Borel theorem
- Function convergence and continuity
- Sequences and limits of sequences
- Sequences and their limits
- Correspondence between function limits and sequences
Outline
1. \( f \) is continuous with a closed interval domain ⇒\( f \) is bounded. (by: continuous functions on a closed interval are bounded)
(proof for this lemma: proof by contradiction—identify an unbounded sub-sequence, then show that it contradicts with Heine-Borel).
2. \( a < b \) ⇒ \( [a, b] \) is non-empty ⇒ \( \{f(x) : x \in [a, b] \} \) is non-empty ⇒
3. \( m := \sup(\{f(x) : x \in [a, b] \}) \) exists.
4. Define a sequence: \( S := (a_n)_{n=1}^{\infty} \) where \( a_n := \text{ some x such that } m - \frac{1}{n} < f(x) \) (requires axiom of choice)
5. There must exist a subsequence of \( S \), \( (a_{n_j})_{j=1}^{\infty} \), that converges to an element of \( [a, b] \) (by Heine-Borel theorem).
6. If the subsequence converges to \( c \in [a, b] \). What is the value of \( f(c) \)? We should take a limit of the sequence \((f(a_{n_j}))_{j=1}^{\infty} \). To do so, view the elements:
\( f(a_{n_j}) > m - \frac{1}{n_j} \ge m - \frac{1}{j} \)
(I find it easier to conceptualize the above by replacing the inequality with \( f(a_{n_j}) = m - \frac{1}{j} + \delta \), for some \( \delta > 0 \))
7. \( \lim_{j \to \infty} f(a_{n_j}) = \lim_{j \to \infty} m - \frac{1}{j} + \delta = m + \delta = m \text{ (the only valid case is for } \delta = 0 \text{)} \)
8. Thus, we have a sequence of elements from \( [a, b] \) that converges to an element \( c \in [a, b] \) and that \( f(c) = m \), the maximum of all values of \( f \) in \( [a, b] \).
Source: p236-237

Title: The intermediate value theorem
Q: Continuous functions whose domain is closed enjoy two useful properties:
- the maximum principle
- the intermediate value theorem
This card covers the second.
Intermediate value theorem (my version)
Let \( a < b \) be reals, let \( X = [a, b] \), and let \( f: X \to \mathbb{R} \) be a continuous function.
Then for every \( y \in [f_{min}, f_{max}] \), where \( f_{min} \) and \( f_{max} \) are the minimum and maximum obtained by \( f \) (which exist by the maximum principle), there is [...] such that [...].
A: an \( x \in [a, b] \); \( f(x) = y \)
BackOnlyText: Intermediate value theorem (standard version)
Let \( a < b \) be reals, let \( X = [a, b] \), and let \( f: X \to \mathbb{R} \) be a continuous function. Let \( y \) be a real between \( f(a) \) and \( f(b) \), in other words, \( min(f(a), f(b)) \ge y \ge max(f(a), f(b)) \).
Then there exists a \( c \in [a, b] \) such that \( f(c) = y \).
Proof: TODO
Source: p239-240

Title: Monotonic functions
Let \( X \subseteq \mathbb{R} \) and \( f : X \to \mathbb{R} \) be a function.
We say that \( f \) is monotone increasing iff \( f(y) \ge f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is strictly monotone increasing iff \( f(y) > f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is monotone decreasing iff \( f(y) \le f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is strictly monotone decreasing iff \( f(y) < f(x) \) whenever \( x,y \in X \) and \( y > x \).
We say that \( f \) is monotone iff \( f \) is monotone increasing or monotone decreasing.
We say that \( f \) is strictly monotone iff \( f \) is strictly monotone increasing or strictly monotone decreasing.
Some properties of monotic functions
- Function continuity implies monotonicity? No
- Function monotinicity implies continuity? No
- Monotone functions on a closed interval obey the maximum principle (with continuity requirement ignored)? Yes
- Monotone functions on a closed interval obey the intermediate value principle (with continuity requirement ignored)? No
- If a function is strictly monotone and continuous, then one very nice property is that it is invertible.
BackOnlyText: Strictly monotone and continuous implies invertable
Let \( a < b \) be real numbers, and let \( f: [a, b] \to \mathbb{R} \) be a function which is both continuous and strictly monotone increasing.
Then \( f \) is a bijection from \( [a, b] \) to \( [f(a), f(b)] \), and the inverse \( f^{-1}: [f(a), f(b)] \to [a, b] \) is also continuous and strictly monotone increasing.
The same can be said replacing 'increasing' with 'decreasing'.
Example: TODO
Source: p242

Title: Uniform continuity
Q: Continuous functions, open intervals and uniform continuity
Card 1 of 3: Uniform continuity
For a continuous function on a closed interval \( [a, b] \) we know it must be [...]; other nice properties follow. If we relax the restriction and consider a continuous function on an open interval, \( (a, b) \), we cannot imply that it is bounded, and we lose subsequence nice properties. Consider \( f: (0, 2) \to \mathbb{R} \) with \( f(x):=\frac{1}{x} \) for an illustration.
Measure of stability
We try to rebuild some new structure around functions on an open interval. The plan is to classify and distinguish continuous functions based on a measure of stability: measure the degree to which they vary over a certain sized interval of their domain.
\( \delta \)-islands of stability
For a function \( f : X \to \mathbb{R} \) to be continuous at \( x_0 \in X \), for any \( \varepsilon > 0 \) there needed to exist some \( \delta > 0 \) such that for all \(x \in (x_0 - \delta, x_0 + \delta)\) we have \( |f(x)-f(x_0)| < \varepsilon \). The \( \delta \) was free to change for both every \( \varepsilon \) and for every \( x_0 \).
Restricting \( \delta \) to [...] we can describe what it means for a function to be uniformly convergent over it's full domain.
Uniform continuity
Let \( X \subseteq \mathbb{R} \) be a set, and let \( f : X \to \mathbb{R} \) be a continuous function. We say that \( f \) is uniformly continuous iff for every real \( \varepsilon > 0 \), there exists a real \( \delta > 0 \) such that \( |f(x_1) - f(x_2)| < \varepsilon \) whenever \( x_1, x_2 \in X \) and \( |x_1 - x_2 | < \delta \).
Tao uses an if rather than iff relationship in his definition, but I don't see why.
A: bounded; be a single value for any \( x_0 \in X \)
BackOnlyText: Continuity vs. uniform continuity—the difference of \( \delta \)
A function \( f : X \to \mathbb{R} \) converges at \( x_0 \) on \( E \) if for every \( \varepsilon > 0 \) there exists a \( \delta > 0 \) such that for all \( x \in \{ i \in X \cap E : |i - x_0 | < \delta \} \) we have \( |f(x) - L| < \varepsilon \) for some real \( L \). \( f \) is continuous at \( x_0 \) if it converges to \( f(x_0) \) at \( x_0 \) on \( X \).
We can see that every uniformly continuous function is continuous but not the other way around.
Example: The function \( f : (0, 2) \to \mathbb{R} \) defined by \( f(x) := \frac{1}{x} \) is continuous on \( (0, 2) \), but not uniformly continuous; the dependency of \( \varepsilon \) on \( \delta \) prescribed by continuity 'becomes worse' as \( x \to 0 \).
Source: p244-246

Title: Uniform continuity (sequence dual)
Uniform continuity: 2/3
ε-δ vs. sequence formulation
Set adherence and function convergence have both an "epsilon-delta" formulation and a sequence formulation. We covered propositions that mapped between these equivalences.
Uniform convergence also has this duality. Below, uniform convergence is formulated in terms of sequences. The sequences equivalence formulations below look very similar to the formulation of equivalent Cauchy sequences, but they are not necessarily Cauchy, and that is their distinction.
Equivalent sequences
Let \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) be sequences of reals. We say that \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) are equivalent sequences iff for any real \( \varepsilon > 0 \) there exists an integer \( N \ge m \) such that \( |a_k - b_k| < \varepsilon \) for all \( k \ge N \).
This formulation can also be split up by using intermediate definitions for ε-close sequences and eventual ε-close sequences.
The above definition can be succinctly formulated using sequence limits also.
Equivalent sequences (via limits)
Let \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) be sequences of reals. We say that \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) are equivalent sequences iff \( \lim_{n \to \infty}(a_n - b_n) = 0\).
Now on to uniform convergence phrased using equivalent sequences.
Uniform convergence (via equivalent sequences)
Let \( X \subseteq \mathbb{R} \) be a set and let \( f: X \to \mathbb{R} \) be a function. Then the following two statements are logically equivalent:
- f is uniformly continuous.
- For any two equivalent sequences \( (a_n)_{n=m}^{\infty} \) and \( (b_n)_{n=m}^{\infty} \) whose elements come from \( X \), the sequences \( (f(a_n))_{n=m}^{\infty} \) and \( (f(b_n))_{n=m}^{\infty} \) are also equivalent sequences.
Useful properties
Three useful properties concerning uniformly continuous functions are:
- they map Cauchy sequences to Cauchy sequences.
- they map bounded sets to bounded sets (the image of a bounded set under a uniformly continuous function is a bounded set).
- If a fuction \( f \) has a closed interval domain and is continuous then it is also uniformly continuous.
The last of these is proved by Tao; the others are exercises.
Mapping of sequences
Function convergence and sequence convergence
The duality of function convergence and sequence convergence means that if \( f \) is a continuous function, then \( f \) maps convergent sequences to convergent sequences.
Uniform function convergence and pairs of equivalent sequences
The duality of uniform function convergence and pairs of equivalent sequences means that if \( f \) is a uniformly convergent function, then \( f \) maps pairs of equivalent sequences to pairs of equivalent sequences.
Connecting the two
To see the connection between the two ideas above, observer that a sequence \( (a_n)_{n=0}^{\infty} \) converges to \( x \) if and only if the sequence \( (a_n)_{n=0}^{\infty} \) is equivalent to \( (x)_{n=0}^{\infty} \).
Example: \( f(x) := \frac{1}{x} \)
Consider the function \( f : (0, 2) \to \mathbb{R} \) defined by \( f(x):= \frac{1}{x} \). We can confirm that the two sequences \( (\frac{1}{n})_{n=1}^{\infty} \) and \( (\frac{1}{2n})_{n=1}^{\infty} \) are equivalent sequences in \( (0, 2) \). However, the sequences \( (f(\frac{1}{n}))_{n=1}^{\infty} \) and \( (f(\frac{1}{2n}))_{n=1}^{\infty} \) (which are \( (n)_{n=1}^{\infty} \) and \( (2n)_{n=1}^{\infty} \) respectively) are not equivalent sequences, so \( f \) cannot be uniformly continuous.
\( f(x) := x^2 \)
The function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x):= x^2 \) is a continuous function, but, it turns out, it is not uniformly continuous. In some sense, the continuity gets worse approaching infinity.
To confirm this, consider the sequences \( (n)_{n=1}^{\infty} \) and \( (n + \frac{1}{n})_{n=1}^{\infty} \). They are equivalent; however, the sequences \( (f(n))_{n=1}^{\infty} \) and \( (f(n + \frac{1}{n}))_{n=1}^{\infty} \) are not. Evaluating \( f \), the sequences become \( (n^2)_{n=1}^{\infty} \) and \( (n^2 + 2 + \frac{1}{n})_{n=1}^{\infty} \) respectively. The former, \( (n^2)_{n=1}^{\infty} \), never becomes 2-close to the latter, \( (n^2 + 2 + \frac{1}{n})_{n=1}^{\infty} \). Thus, it can be concluded that \( f \) is not uniformly continuous.
Source: 246-247

Title: Limits at infinity (for continuous function)
Formulations of the limit \( \lim_{x \to x_0; x \in X \cap E}f(x) \) for a function \( f : X \to \mathbb{R} \), where \( E \subseteq X \subseteq \mathbb{R} \), so far have covered the case where \( x \to x_0 \) where \( x_0 \) is a real number. Below, the idea is extended to describe what it means for limits of \( f \) when \( x_0 \) equals \( +\infty \) or \( -\infty \).
Infinite adherent points
Let \( X \subseteq \mathbb{R} \).
We say that \( +\infty \) is adherent to \( X \) iff for every \( M \in X \) there exists an \( x \in X \) such that \( x > M \).
We say that \( -\infty \) is adherent to \( X \) iff for every \( M \in X \) there exists an \( x \in X \) such that \( x < M \).
In other words, \( +\infty \) is adherent to \( X \) iff \( X \) has no upper bound, or equivalently, \( \sup(X) = +\infty \). Similarly, \( -\infty \) is adherent to \( X \) iff \( X \) has no lower bound, or equivalently, \( inf(X) = -\infty \).
So a set is bounded iff \( +\infty \) and \( -\infty \) are not adherent points.
Limits at infinity
Let \( X \subseteq \mathbb{R} \) with \( +\infty \) being an adherent point, and let \( f: X \to \mathbb{R} \) be a function.
We say that \( f \) converges to \( L \) as \( x \to +\infty \) in \( X \), and write \( \lim_{x \to +\infty} f(x) = L \) iff
for any real \( \varepsilon > 0 \) there exists an \( M \in X \) such that \( |f(x) - L| \le \varepsilon \) for all \( x \in X \text{ such that } x > M \).
A similar formulation can be made for \( x \to -\infty \).
Source: p250-251