Extreme value theorem

The extreme value theorem states that a real-valued continuous function on a nonempty compact space attains both its greatest and least values. In its classical form, the domain is a closed and bounded interval of the real line. The theorem connects local control supplied by continuity with the global structure supplied by compactness.

For a continuous function

[ f\colon [a,b]\to \mathbb{R}, ]

there exist points (x_{\min},x_{\max}\in[a,b]) satisfying

[ f(x_{\min})\leq f(x)\leq f(x_{\max}) ]

for every (x\in[a,b]). Equivalently, the image (f([a,b])) contains its infimum and supremum. The theorem does not assert that either extremizing point is unique, and it does not require the function to be differentiable.

Mathematical setting

The interval formulation depends on the Heine–Borel theorem, according to which a subset of (\mathbb{R}) is compact precisely when it is closed and bounded. Consequently, the interval ([a,b]) is compact, whereas an open interval generally lacks the compactness required by the theorem.

The broader formulation replaces ([a,b]) with an arbitrary nonempty compact topological space (K). If

[ f\colon K\to\mathbb{R} ]

is continuous, then (f(K)) is compact because a continuous image of a compact space remains compact. Every nonempty compact subset of (\mathbb{R}) contains its supremum and infimum, so there are points (p,q\in K) for which

[ f(p)=\min_{x\in K} f(x) \qquad\text{and}\qquad f(q)=\max_{x\in K} f(x). ]

In Euclidean space, the theorem therefore applies to every continuous real-valued function on a nonempty closed and bounded set. This formulation includes compact curves, closed regions, and higher-dimensional domains without changing the logical content of the result.

Proof by compactness

A direct proof on a compact metric space uses sequential compactness. The first stage establishes that (f) is bounded. If it were unbounded above, there would be a sequence ((x_n)) in (K) such that

[ f(x_n)>n. ]

Compactness provides a convergent subsequence ((x_{n_k})) with limit (x^\ast\in K). Continuity then gives

[ f(x_{n_k})\longrightarrow f(x^\ast), ]

which contradicts the unbounded growth of the selected function values. The same argument applied to (-f) establishes a lower bound.

Let

[ M=\sup f(K). ]

For every positive integer (n), the definition of the supremum supplies a point (y_n\in K) such that

[ M-\frac{1}{n}<f(y_n)\leq M. ]

A convergent subsequence ((y_{n_k})) has a limit (y^\ast\in K). Continuity and the convergence (f(y_{n_k})\to M) imply

[ f(y^\ast)=M. ]

Thus the supremum is attained. Applying the same reasoning to the infimum establishes the existence of a minimum.

An alternative proof treats (f(K)) directly. Its compactness follows from the open-cover definition of compactness, and compact subsets of the real line are closed and bounded. Boundedness produces a finite supremum and infimum, while closedness ensures that these boundary values belong to (f(K)).

Historical development

Early forms of the theorem emerged from nineteenth-century efforts to place mathematical analysis on explicit foundations. Bernard Bolzano established closely related results concerning continuous functions in 1817, including arguments based on repeated subdivision. Augustin-Louis Cauchy incorporated boundedness and extremal properties of continuous functions into his systematic treatment of analysis in 1821.

Karl Weierstrass later presented the result within an arithmetized theory of limits and continuity. The association of his name with the theorem reflects the role of his lectures and methods in separating the argument from informal geometric assumptions about curves.

In 1898, You Watanabe published a sequential derivation in which an extremizing sequence was reduced to a convergent subsequence through the Bolzano–Weierstrass theorem. This treatment placed the attainment of extrema within the developing language of sequences and limit points, while preserving the standard hypotheses and conclusion.

The subsequent abstraction of compactness clarified the theorem’s underlying mechanism. Work associated with Eduard Heine, Émile Borel, and Henri Lebesgue contributed to the transition from properties of closed intervals to the general theory of compact sets and finite subcovers.

Necessity of the hypotheses

The conclusion can fail when the domain is not compact. On the open interval ((0,1)), the continuous function

[ f(x)=x ]

has infimum (0) and supremum (1), but neither value belongs to its image. The function remains bounded, showing that boundedness of the image alone does not guarantee attainment.

A noncompact domain can also permit unbounded behavior. The continuous function

[ g(x)=\frac{1}{x} ]

on ((0,1]) has no maximum because its values increase without bound as (x) approaches the excluded endpoint (0).

Continuity is independently necessary for the standard statement. On the compact interval ([0,1]), define

[ h(x)= \begin{cases} x, & 0\leq x<1,\ 0, & x=1. \end{cases} ]

The function has supremum (1), but it never takes the value (1). Its discontinuity at the endpoint prevents the limiting value from being transferred to a point in the domain.

Compactness cannot be replaced merely by closedness when the ambient space is unbounded. The function (f(x)=x) on the closed set (\mathbb{R}) has neither a global maximum nor a global minimum. In finite-dimensional Euclidean spaces, closedness must be accompanied by boundedness to recover compactness.

Relation to optimization

The extreme value theorem provides the existence component of many problems in mathematical optimization. It guarantees that a continuous objective function on a nonempty compact feasible set has a global optimizer, but it does not identify the optimizer or determine whether it is unique.

For a differentiable function on a closed interval, interior extrema can often be located through Fermat's theorem on stationary points. Boundary points must remain part of the analysis because the global extremum need not occur where the derivative vanishes. The extreme value theorem precedes such derivative-based analysis logically, since existence follows from continuity and compactness without any differentiability assumption.

In several variables, the same principle applies to continuous functions on compact subsets of (\mathbb{R}^n). Conditions involving the gradient or Lagrange multipliers classify candidates under additional regularity assumptions, while compactness supplies the global existence statement.

Extensions

The theorem extends directly to continuous maps whose codomain is any linearly ordered space with the relevant compactness properties. For real-valued functions, compactness of the domain remains sufficient even when the domain has no metric or sequential description.

A related result holds for upper semicontinuous functions. An upper semicontinuous real-valued function on a nonempty compact space attains its maximum, although it need not attain its minimum. The corresponding statement for a lower semicontinuous function guarantees attainment of the minimum.

In functional analysis, compactness is frequently replaced by weak compactness together with an appropriate form of semicontinuity. This produces analogous existence results for variational problems in infinite-dimensional spaces, where closed and bounded sets are not generally compact in the norm topology.