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

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.


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