Theorem

Extreme Value Theorem

Analysis

The Extreme Value Theorem states that a continuous real-valued function on a closed, bounded interval attains both a maximum value and a minimum value somewhere on that interval. It is a basic existence result in real analysis, guaranteeing that optimization problems posed over compact domains have solutions, and its proof depends on the completeness of the real numbers.

Facts

Partially Attested

Proof Year

1830 1

Bolzano proved this in the 1830s in an unpublished manuscript, Function Theory, not printed until 1930; Karl Weierstrass later gave the formulation most textbooks now cite, so no single settled proof year is attested.

Statement

A continuous real valued function on a closed and bounded interval attains both a maximum value and a minimum value somewhere on that interval. 1

Classification

Statement Form

Existence Theorem 1

Connections

Sources

1. Extreme Value Theorem (Wikipedia)

Wikimedia Foundation

History section
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.
History section, second sentence
Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.

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Extreme Value Theorem

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