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 $$x_{min}$$).