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

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 [...].

We say that \( f \) is bounded from below iff [...].

We say that \( f \) is bounded iff it is both bounded from above and bounded from below.

[...] are bounded

Let \( a < b \) be real numbers. Let \( f : [a,b] \to \mathbb{R} \) be a function [...] 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 \( x_{min} \)).