Extreme Value Theorem¶

Theorem¶

Let $a<b$ be real numbers, and let $f\colon [a, b] \to\mathbb{R}$ be a continuous function on $[a, b]\text{.}$ Then $f$ attains its global maximum at $x=x_{\mathrm{max}} \in [a, b]\text{,}$ and also attains its global minimum at $x=x_{\mathrm{min}} \in [a, b]\text{.}$

Interpretation¶

The extreme value theorem guarantees that a continuous function on a closed and bounded interval attains both a largest and a smallest value. It does not say where they are, only that a search for them cannot come back empty.

A continuous function on a closed interval from a to b with its maximum attained inside the interval and its minimum attained at the endpoint b.

In the figure the maximum is attained at the interior point $c$ , the highest point of the graph, and the minimum at the endpoint $b$ . Both are values that $f$ actually takes, $f(c)$ and $f(b)$ , not merely bounds that it approaches. The theorem allows either kind of location, so the endpoints always have to be checked.

On a closed interval both extreme values are attained. Switch to the open interval $(a,b)$ , or to the function with a jump, and watch an extreme value turn into a supremum or infimum that is never reached.

Each hypothesis is needed. On the open interval $(0,1)$ the function $f(x)=x$ comes arbitrarily close to $1$ without reaching it, so it has no maximum; on the unbounded interval $[0,+\infty)$ the same function is not even bounded above. Without continuity a jump can remove the extreme value: on $[0,1]$ , the function equal to $x$ for $x<1$ and to $0$ at $x=1$ has supremum $1$ but never takes the value $1$ . The hypotheses are sufficient, not necessary: $\sin x$ attains its maximum $1$ on the open interval $(0,\pi)$ , at $x=\tfrac{\pi}{2}$ $\frac{π}{2}$ .

In the language of global extrema, the theorem says that a continuous $f$ on $[a,b]$ has a global maximum and a global minimum; in particular, $f$ is bounded there and its supremum is a maximum. Math III extends the theorem from closed bounded intervals to a wider class of sets in the extreme value theorem on compact sets.

Key Results¶

The theorem does not apply to open or unbounded intervals.
The theorem does not apply to functions that are not continuous.

Examples¶

The function $f(x)=x^2$ on $[0,1]$ attains the largest value $1$ and the smallest value $0\text{.}$
The function $f(x)=x$ has no maximum on $(0,1)\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 5

Aliases: EVT, Weierstrass extreme value theorem Symbols: element of, infinity, pi, real numbers

Last updated: 20 October 2026 03:02