Theorems
Intermediate Value Theorem
Also Known As IVT
Analysis
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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.
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 Cross-Tradition Connections
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Proved By
Cauchy gave the modern formulation independently, a few years after Bolzano's original proof.
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