Title: Math and science / Analysis / Tao / 10: Differentiation of functions

Title: Differentiability at a point, definition
Q: Differentiability at a point
Let \( X \) be a subset of \( \mathbb{R} \), and let \( x_0 \in X \) be an element of \( X \) that is also a limit point of \( X \). Let \( f : X \to \mathbb{R} \) be a function. Then, if the following limit:
[...]
converges to some \( L \) then we say that \( f \) is differentiable at \( x_0 \) on \( X \) with derivative \( L \). We write: \( f'(x_0) := L \).
If \( x_0 \) is not an element of \( X \) or is not a limit point of \( X \), or the limit does not converge, we leave \( f'(x_0) \) undefined and we say that \( f \) is not differentiable at \( x_0 \) on \( X \).
We need \( x_0 \) to be a limit point (not isolated) in order for it to be adherent to \( X \setminus \{x_0\} \) and in turn for the above limit to be defined. Thus, functions do not have a derivative defined at isolated points.
A: \[ \lim_{x \to x_0; x \in X \setminus \{x_0\} } \frac{f(x) - f(x_0)}{x - x_0} \]
BackOnlyText: Reasons against the notation \( \frac{df}{dx} \)
Consider two functions \( f : X \to \mathbb{R} \), defined by \( f(x) := x^2 \) and \( g : X \to \mathbb{R} \), defined by \( g(y) := y^2 \). We have \( f = g \). If we write \( \frac{df}{dx} \) instead of \( f' \), then it is troublesome to consider \( \frac{dg}{dx} \), is it 0? Introducing the notion of differentiation along vector fields can resolve this ambiguity, but this is out of scope here.
Example: Let \( f : \mathbb{R} \to \mathbb{R} \) be the function \( f(x) := x^2 \), and let \( x_0 \) be any real number. To see whether \( f \) is differentiable at \( x_0 \) on \( \mathbb{R} \), we compute the limit:
\[ \begin{aligned} \lim_{x \to x_0; x \in X \setminus \{x_0\} } \frac{f(x) - f(x_0)}{x - x_0} &= \lim_{x \to x_0; x \in X \setminus \{x_0\} } \frac{x^2 - x_0^2}{x - x_0} \\
&= \lim_{x \to x_0; x \in X \setminus \{x_0\} } \frac{(x + x_0)(x - x_0)}{x - x_0} \\
&= \lim_{x \to x_0; x \in X \setminus \{x_0\} } (x + x_0) \\
&= 2 x_0 \\
\end{aligned} \]
Thus \( f \) is differentiable at \( x_0 \) on \( \mathbb{R} \) and its derivative is \( 2 x_0 \).
Another example:
Source: p252

Title: Newton's approximation
Q: Informally, if \( f \) is differentiable at \( x_0 \), then one has the approximation \( f(x) \approx f(x_0) + f'(x_0)(x - x_0) \), and conversely.
The formal version:
Newton's approximation
Let \( X \) be a subset of \( \mathbb{R} \), let \( x_0 \in X \) be a limit point of \( X \), let \( f : X \to \mathbb{R} \) be a function, and let \( L \) be a real number. Then the following statements are logically equivalent:
1. \( f \) is differentiable at \( x_0 \) with derivative \( f' \).
2. For any \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \in \{i \in X : |i - x_0| < \delta \} \), \( f(x) \) is [...]-close to [...].
In other words, we have:
\( |f(x) - (f(x_0) + f'(x_0)(x - x_0))| \le \varepsilon|x - x_0| \)
2. can be prased as: fix \( x_0 \). Given a challenge of \( \varepsilon \), it's possible to find a neighbourhood around \( x_0 \) for which \( x_0 \) along with \(f(x_0) \) and \(f'(x_0) \) can be used to approximate \( f \) for all of the neighbourhood, with an approximation error at most \( \varepsilon \).
A: \( \varepsilon |x - x_0| \); \( f(x_0) + f'(x_0)(x-x_0) \)
BackOnlyText: This seems like such a confusing way to formulate the idea.
An attempt at the formal version in words as a challenge: for any \( x_0 \) at which \( f \) is differentiable you may choose any \( \varepsilon > 0 \) and I can can give you a \( \delta > 0 \) such that any point that is within \( \delta \) from \( x_0 \), an estimate of \( f(x_0) + f'(x_0)(x - x_0) \) will be at most \( \delta \varepsilon \) off. And in fact, I can do slightly better, the error will be within \( |x - x_0| \varepsilon \).
Source: p255

Title: Differentiability on a domain
Q: Let \( X \) be a subset of \( \mathbb{R} \) and let \( f : X \to \mathbb{R} \) be a function. We say that \( f \) is differentiable on \( X \) iff [...].
Tao describes this as an if rather than iff statement.
A: for every limit point \( x_0 \in X \), \( f \) is differentiable at \( x_0 \) on \( X \)
BackOnlyText: I think this definition is allowing isolated points to be ignored. Note how the limit point is also an element, so we are also excluding points that are limit points but not in the \( X \).
Source: p255

Title: Differentiability ⇒ continuity
Q: Differentiability implies continuity
Let \( X \) be a subset of \( \mathbb{R} \), and let \( f : X \to \mathbb{R} \) be a function. Let \( x_0 \) be [...]. If \( f \) is differentiable at \( x_0 \) on \( X \), then \( f \) is continuous at \( x_0 \).
Tao's definition of function continuity includes the assumption that the limit is being taken over \( X \), hence why 'on \( X \) is missing from the end of the proposition above.
The definition of differentiation on a domain along with the proposition above that differentiability implies continuity brings us to an immediate corollary.
Let \( X \) be a subset of \( \mathbb{R} \) and let \( f : X \to \mathbb{R} \) be a function. If \( f \) is differentiable on \( X \), then \( f \) is continuous.
I'm not quite sure why Tao decides to now include the idea of 'continuity on X'.
A: an element of X and a limit point of \( X \)
Source: p255

Title: Rolle's theorem
Q: Rolle's theorem
Let \( a < b \) be real numbers, and let \( g : [a, b] \to \mathbb{R} \) be a function such that:
1. \( g \) is [something and something].
2. [an equality].
Then there exists an \( x \in (a, b) \) such that \( g'(x) = 0 \).
A corollary of Rolle's theorem is the [something] theorem.
A: continuous on \( [a, b] \) and differentiable on \( (a, b) \); \( g(a) = g(b) \); mean value
BackOnlyText: I think the following concepts are used when proving Rolle's theorem.
Local maxima and minima
Let \( f : X \to \mathbb{R} \) be a function and let \( x_0 \in X \). We say that \( f \) obtains a local maxima at \( x_0 \) if there exists some \( \delta > 0 \) such that the restricted function \( f |_{X \cap (x_0 - \delta, x_0 + \delta)} \) of \( f \) obtains a maximum at \( x_0 \).
Local minima are defined similarly.
Local extrema are stationary
Let \( a < b \) be real numbers, let \( x_0 \in (a, b) \), and let \( f : (a, b) \to \mathbb{R} \) be a function which is differentiable at \( x_0 \). If \( f \) attains either a local maximum or local minimum at \( x_0 \), then \( f'(x_0) = 0 \).
Source: p260

Title: Mean value theorem
Q: Let \( a < b \) be real numbers, and let \( g : [a, b] \to \mathbb{R} \) be a function [something and something]. Then, there exists an \( x \in (a, b) \) such that: [\( g'(x) = \) ?]
A: continuous on \( [a, b] \) differentiable on \( (a, b) \); \[ g'(x) = \frac{g(b) - g(a)}{b - a} \]
BackOnlyText: The mean value theorem is a corollary of Rolle's theorem.
Source: p260

Title: Derivative of monotone functions
Q: Monotone increasing implies [...]
Let \( X \) be a subset of \( \mathbb{R} \), and let \( x_0 \in X \) be a limit point of \( X \). Let \( f: X \to \mathbb{R} \) be a function. If \( f \) is [...] and is [...], then \( f'(x_0) \ge 0 \).
If instead, \( f \) is monotone decreasing, then \( f'(x_0) \le 0 \).
A: a non-negative derivative; monotone increasing; differentiable at \( x_0 \)
BackOnlyText: Strictly monotone doesn't imply strict derivative
Strictly monotone increasing/decreasing does not imply a strictly positive/negative derivative.
Strict derivative implies strictly monotone
We do have the result that if a function has a strictly positive derivative, then it must be strictly monotone increasing.
What about non-negative derivative implying monotone increasing? Tao doesn't quite explain if this is true or not. If it is true, you might have to include differentiability of the function on the whole domain.
Source: p261