Intermediate value theorem
The intermediate value theorem states that a continuous real-valued function on a closed interval takes every value between its endpoint values. In particular, if its values at the endpoints have opposite signs, it must have a zero between them. The theorem guarantees existence, not the location or number of solutions, and underlies methods such as interval bisection for approximating roots.
Statement
Let f be a real-valued function continuous on the closed interval [a, b]. If m lies between f(a) and f(b), then there is at least one c in [a, b] such that f(c) = m. When m lies strictly between the two endpoint values, c can be chosen in the open interval (a, b). Continuity is essential: the theorem rules out a jump across an intermediate value.
Geometric meaning
A continuous graph joining the points (a, f(a)) and (b, f(b)) cannot skip a height between f(a) and f(b). The theorem says that the graph reaches every such intermediate height at least once. It does not say that the function is monotonic, that the value is reached only once, or where the corresponding input c is located.
Existence of roots
A useful special case takes m = 0. If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then 0 lies between the endpoint values, so some c in (a, b) satisfies f(c) = 0. This proves that an equation has a solution in the interval without necessarily finding its exact value. For instance, a continuous polynomial whose values at two endpoints have opposite signs must have a real root between them.
Applications and limits
The theorem gives a foundation for the bisection method: starting with a continuous function whose endpoint values have opposite signs, repeatedly keep a half-interval whose endpoint values still bracket zero. The retained intervals shrink around a root, yielding successively closer approximations. The intermediate value theorem itself only guarantees existence; it does not provide a numerical approximation, prove uniqueness, or determine how many roots occur. Those questions require additional arguments or methods.
Key facts
- Hypothesis
- The function must be continuous on the entire closed interval [a, b].
- Conclusion
- Every value between f(a) and f(b) is attained somewhere on [a, b].
- Root test
- Opposite signs at the endpoints guarantee at least one zero in the open interval.
- What it does not show
- The theorem does not locate the solution, establish uniqueness, or count all solutions.
- Numerical connection
- The bisection method repeatedly narrows an interval that brackets a zero, relying on continuity.
Ask about this article
Answers use only “Intermediate value theorem” and its references, with citations.
References
- 1.2.4 Continuity
OpenStax, Rice University · other · reliability: unverified · retrieved 30 October 2026
Supports: Introduces continuity and the intermediate value theorem, with examples and its use to establish existence of roots.
- 2.Continuity and the Intermediate Value Theorem
University of Texas at Austin, Department of Mathematics · other · reliability: unverified · retrieved 30 October 2026
Supports: States the intermediate value theorem and illustrates how it proves existence of polynomial roots without computing them.
- 3.Calculus I – Continuity
Paul's Online Math Notes, Lamar University · other · reliability: unverified · retrieved 30 October 2026
Supports: Explains the theorem's hypotheses, intermediate-value interpretation, root-existence application, and limits on what it establishes.
- 4.Intermediate Value Theorem Problems
University of California, Davis, Department of Mathematics · other · reliability: unverified · retrieved 30 October 2026
Supports: States the theorem, presents its existence-only use for equations, and gives examples of root and fixed-point applications.
- 5.2.1 The Bisection Method
University of Notre Dame, Department of Applied and Computational Mathematics and Statistics · other · reliability: unverified · retrieved 30 October 2026
Supports: Shows how the intermediate value theorem guarantees a root in an interval with opposite-sign endpoint values and motivates repeated bisection to approximate it.
- 6.Intermediate Value Theorem
ProofWiki · other · reliability: unverified · retrieved 30 October 2026
Supports: States the theorem and identifies Bolzano's theorem as an alternate name used by some sources.
References are suggested by the research engine and may be incomplete. Unverified URLs are omitted rather than guessed.
Article history
- Version 130 October 2026 · EditorCreated by Zapia