The intermediate value theorem states that if a continuous function's domain contains an interval [a, b] and s is a number strictly between f(a) and f(b), then there exists some point x between a and b where f(x) equals s. Bernard Bolzano gave the first rigorous proof in 1817. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementA continuous function defined on a closed interval that takes two different values at its endpoints must take every value between those two endpoint values somewhere inside the interval. 1 Classification
Statement Form Connections
Associated With
Continuity, Concepts The theorem's hypothesis is that the function is continuous on the interval, so its subject is the continuity concept itself.
Additional Source Intermediate Value Theorem (Wikipedia)introductory paragraph
Has Statement Form
In Branch
Source Intermediate Value Theorem (Wikipedia)
Proved By
Cauchy gave the modern formulation independently, a few years after Bolzano's original proof.
Source Intermediate Value Theorem (Wikipedia)
Source Intermediate Value Theorem (Wikipedia)
Sources
1. Intermediate Value Theorem (Wikipedia)
Wikimedia Foundationintroductory paragraph
if f is a continuous function whose domain contains the interval [a, b] and s is a number such that f(a)<s<f(b), then there exists some x between a and b such that f(x)=s
History section
The theorem was first proved by Bernard Bolzano in 1817.
Lead section, statement-form reference
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b] and s is a number such that f ( a ) < s < f ( b ) , then there exists some x between a and b such that f ( x ) = s .
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