The Weierstrass extreme value theorem

Summary

This lesson opens the passage from the pointwise to the global. Continuity at a point controls the function only near that point; the Weierstrass theorem asserts that, if the function is continuous at every point of a closed and bounded interval, it is bounded on the whole interval and attains on it a maximum value and a minimum value. Both assertions are proved with the completeness axiom, through the supremum, like Bolzano’s theorem: boundedness, with the supremum of the points up to which the function is bounded and the local boundedness of the lesson on limit and continuity; the existence of the maximum, with the supremum of the values and an auxiliary function that would be continuous and unbounded if that supremum were not attained. The lesson shows with counterexamples that no hypothesis can be dropped, deduces that the image of a closed and bounded interval is another interval of the same type and that, if the function takes the same value at the endpoints, one of its extrema is attained in the interior, and applies the theorem to positive functions, to periodic functions and to functions defined on the whole line.

Learning objectives

Distinguish the supremum of the values of a function from its maximum value, and express with quantifiers that a function attains its maximum or its minimum on a set.
Prove with the completeness axiom that a function continuous on a closed and bounded interval is bounded and attains its maximum and its minimum.
Show with counterexamples that the theorem fails if the interval is not closed, if it is not bounded or if the function is not continuous, and recognize that its hypotheses are sufficient, not necessary.
Use the theorem, together with the intermediate value theorem and limits at infinity, to find the image of a closed and bounded interval and to prove that a function attains its extrema on domains that are not closed and bounded intervals.

Maximum, minimum and supremum of a function

The two preceding units studied functions at a point: the limit, continuity and the derivative describe what happens near $a$ . Many questions, by contrast, concern a whole set: what is the largest value that a function takes on an interval, if there is one, or at what point it is attained. The function $x$ on $\ioo{0}{1}$ has no value larger than all the others: for each $x$ in the interval, the number $\frac{x + 1}{2}$ also lies in it, and is larger. Its values approach $1$ without reaching it. It is therefore advisable to separate two notions: the least upper bound of the values, which exists as soon as the values are bounded, and the maximum value, which is a value of the function.

DefinitionBounded function; maximum and minimum of a function on a set

Let $f$ be a function defined on a set $A$ and $S$ a nonempty subset of $A$ . We say that $f$ is bounded above on $S$ if there exists a number $K$ such that $f(x) \leq K$ for every $x$ in $S$ ; bounded below, if there exists $K$ such that $f(x) \geq K$ for every $x$ in $S$ ; and bounded, if it is bounded in both ways, that is, if there exists $K$ such that $\abs{f(x)} \leq K$ for every $x$ in $S$ . We say that $f$ attains its maximum on $S$ at a point $x_0$ of $S$ if the value $f(x_0)$ is not exceeded by any other value of $f$ on $S$ , and that it attains its minimum at $x_0$ if there is no smaller value on $S$ :

(\forall x \in S)\left(f(x) \leq f(x_0)\right), \qquad (\forall x \in S)\left(f(x_0) \leq f(x)\right)

The number $f(x_0)$ is the maximum value (or the minimum value) of $f$ on $S$ , and we write $\max_{x \in S} f(x)$ (or $\min_{x \in S} f(x)$ ); the point $x_0$ is a maximum point (or a minimum point).

The maximum value, if it exists, is unique, because two maximum values would each be greater than or equal to the other; the maximum point need not be: the cosine attains its maximum $1$ on $\icc{0}{4\pi}$ at three points, $0$ , $2\pi$ and $4\pi$ . The lesson on maxima and minima will call these extrema absolute, to distinguish them from the local ones, which only compare $f(x_0)$ with the nearby values.

If $f$ is bounded above on $S$ , the set of its values, $f(S)$ , is not empty and is bounded above; by the completeness axiom, recalled in the lesson on the real line, it has a supremum, which is written $\sup_{x \in S} f(x)$ . The relation between the two notions is simple: $f$ attains its maximum on $S$ if and only if the supremum of its values is one of them, and then the two coincide. Indeed, if $f(x_0)$ is the maximum value, it is an upper bound of $f(S)$ that belongs to $f(S)$ , and no smaller number can be an upper bound; conversely, if the supremum is a value $f(x_0)$ , no value exceeds it. The function $x$ on $\ioo{0}{1}$ is bounded, its supremum is $1$ and it does not attain it; its infimum is $0$ and it does not attain it either. The Weierstrass theorem gives conditions under which this does not happen.