Extreme Value Theorem

analysis

Definition

Extreme Value Theorem

Let $f:[a,b]\to \mathbb{R}$ be a continuous function on a closed and bounded interval $[a,b]$. Then $f$ attains both an absolute maximum and an absolute minimum at least once.

That is, there exist points $c,d\in [a,b]$ such that for all $x\in [a,b]$:

$$f(c)\le f(x)\le f(d)$$

Generalisation

The theorem can be generalised to higher dimensions: a continuous function from a compact set to the real numbers is bounded and attains its extrema. In ${\mathbb{R}}^n$, by the Heine-Borel theorem, a set is compact if and only if it is closed and bounded.

Necessity of Conditions

• Continuity: If $f$ is not continuous, it may not attain extrema (e.g., a function with a jump discontinuity at its peak).
• Closed Interval: On an open interval $(a,b)$, a function like $f(x)=x$ does not attain a maximum or minimum.
• Boundedness: On an unbounded set, a function like $f(x)=e^x$ does not attain a maximum.