intermediate value theorem

mathematics

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July 16, 2026

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The intermediate value theorem is a fundamental concept in calculus. It states that if a continuous function f on a closed interval [a, b] has values f(a) and f(b) with opposite signs, then there must be at least one point c within the interval (a, b) where f(c) equals zero. In simpler terms, an unbroken curve must cross the x-axis at least once between two points if those points are on opposite sides of the x-axis. This theorem is closely related to Darboux's theorem, which extends this idea to the values of derivatives. The mean-value theorem, another related concept, states that the slope of a secant line connecting two points on a smooth curve is equal to the slope of a tangent line at some point between those two points.12

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Brouwer’s fixed point theorem
…to be equivalent to the intermediate value theorem, which is a familiar result in calculus and states that if a continuous real-valued function f defined on the closed interval [−1, 1] satisfies f(−1) < 0 and f(1) > 0, then f(x) = 0 for at least one number x between…
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…to be equivalent to the intermediate value theorem, which is a familiar result in calculus and states that if a continuous real-valued function f defined on the closed interval [−1, 1] satisfies f(−1) < 0 and f(1) > 0, then f(x) = 0 for at least one number x between…
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Darboux’s theorem
The intermediate value theorem, which implies Darboux’s theorem when the derivative function is continuous, is a familiar result in calculus that states, in simplest terms, that if a continuous real-valued function f defined on the closed interval [−1, 1] satisfies f(−1) < 0 and f(1) >…
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The intermediate value theorem, which implies Darboux’s theorem when the derivative function is continuous, is a familiar result in calculus that states, in simplest terms, that if a continuous real-valued function f defined on the closed interval [−1, 1] satisfies f(−1) < 0 and f(1) >…
Read More

analysis
Arithmetization of analysis
…weaker assumption known as the intermediate value theorem: if f(x) is a continuous function of a real variable x and if f(a) < 0 and f(b) > 0, then there is a c between a and b such that f(c) = 0.
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…weaker assumption known as the intermediate value theorem: if f(x) is a continuous function of a real variable x and if f(a) < 0 and f(b) > 0, then there is a c between a and b such that f(c) = 0.
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