# Intermediate value theorem
The intermediate value theorem (IVT) states that a real-valued function continuous on a closed interval \([a,b]\) takes every value between \(f(a)\) and \(f(b)\): for any \(y\) strictly between \(f(a)\) and \(f(b)\), there is at least one \(c \in (a,b)\) with \(f(c) = y\)[1](https://unseel.com/mathematics/intermediate-value-theorem). The only hypothesis is continuity; no differentiability, monotonicity, or smoothness is required, and the conclusion is existence only, saying nothing about uniqueness or how to find \(c\)[1](https://unseel.com/mathematics/intermediate-value-theorem).
| Key fact | Detail |
|---|---|
| Statement | If \(f : [a,b] \to \mathbb{R}\) is continuous and \(y\) lies strictly between \(f(a)\) and \(f(b)\), then some \(c \in (a,b)\) satisfies \(f(c) = y\)[1](https://unseel.com/mathematics/intermediate-value-theorem) |
| Root-finding form | If \(f(a)\) and \(f(b)\) are nonzero with opposite signs, some \(p \in (a,b)\) satisfies \(f(p) = 0\)[2](https://www.geneseo.edu/~aguilar/public/assets/courses/345/numerical-analysis.pdf) |
| First proof | Bernard Bolzano, 1817, in *Rein analytischer Beweis*; Augustin-Louis Cauchy proved it independently in 1821[3](https://mathshistory.st-andrews.ac.uk/Biographies/Bolzano/) • [4](https://encyclopediaofmath.org/wiki/Arithmetization_of_analysis) |
| Dependence | The theorem rests on [completeness of the real numbers](https://www.edgepedia.org/completeness-of-the-real-numbers); on the rationals it fails, e.g. \(f(x) = x^2 - 2\) takes \(-1\) at \(x=1\) and \(+2\) at \(x=2\) but never \(0\), since \(\sqrt{2}\) is irrational[1](https://unseel.com/mathematics/intermediate-value-theorem) |
| Topological form | The continuous image of a connected set is connected, so \(f([a,b])\) is an interval containing every value between \(f(a)\) and \(f(b)\)[5](https://mathworld.wolfram.com/IntermediateValueTheorem.html) |
| Algorithmic use | The bisection method applies the theorem repeatedly; after \(n\) steps the error satisfies \(\lvert p - p_n \rvert < (b-a)/2^n\)[2](https://www.geneseo.edu/~aguilar/public/assets/courses/345/numerical-analysis.pdf) |
| Constructive status | The classical statement entails the lesser limited principle of omniscience and is not constructively provable as stated, though weaker constructive versions hold[6](https://plato.stanford.edu/entries/mathematics-constructive/) |
## Statement and hypotheses
Let \(f\) be continuous on the closed, bounded interval \([a,b]\). If \(z\) is any real number between \(f(a)\) and \(f(b)\), then there is a number \(c\) in \([a,b]\) satisfying \(f(c) = z\)[7](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/02%3A_Limits/2.04%3A_Continuity). Continuity on a closed interval means continuity at every interior point together with one-sided continuity at the endpoints[7](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/02%3A_Limits/2.04%3A_Continuity).
The hypotheses matter in both directions. A function that is not continuous on the whole interval can evade the conclusion: for \(f(x) = 1/x\), \(f(-1) = -1 < 0\) and \(f(1) = 1 > 0\), yet \(f\) has no zero in \([-1,1]\) because it is not continuous there[7](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/02%3A_Limits/2.04%3A_Continuity). Conversely, the theorem gives no information when the endpoint values have the same sign: \(f(x) = (x-1)^2\) satisfies \(f(0) = 1\) and \(f(2) = 1\) but still has a zero at \(x = 1\), so matching signs do not rule out roots[7](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/02%3A_Limits/2.04%3A_Continuity).
## Why completeness is essential
The theorem feels obvious because the real numbers are constructed to make it true. The completeness axiom, under which every bounded set of reals has a least upper bound, is exactly the property that prevents a continuous curve from skipping a horizontal line[1](https://unseel.com/mathematics/intermediate-value-theorem).
On the rationals, where completeness fails, the theorem is false. As an illustration of how the theorem fails over the rationals, where completeness is lacking, consider \(f(x) = x^2 - 2\): this continuous function gives \(-1\) at \(x = 1\) and \(2\) at \(x = 2\), yet it never equals \(0\) at any rational input, since \(\sqrt{2}\) is irrational[1](https://unseel.com/mathematics/intermediate-value-theorem). This example shows that the theorem is a statement about the number system as much as about functions.
## Proof ideas
The standard proof uses bisection together with the nested interval property. Assume \(f(a) \le v \le f(b)\). Set \(x_1 = a\) and \(y_1 = b\), and let \(m_1\) be the midpoint. If \(f(m_1) \le v\), relabel \(x_2 = m_1\), \(y_2 = y_1\); otherwise relabel \(x_2 = x_1\), \(y_2 = m_1\). Repeating produces two sequences with \(x_n \le c \le y_n\) for a common limit point \(c\), interval lengths halving at each step, and \(f(x_n) \le v \le f(y_n)\) for every \(n\). Continuity of \(f\) at \(c\) then forces \(f(c) = v\)[8](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/07%3A_Intermediate_and_Extreme_Values/7.02%3A_Proof_of_the_Intermediate_Value_Theorem).
The topological proof is shorter: \(f([a,b])\) is connected because the image of a connected set under a continuous function is connected, and since \(c\) lies between \(f(a)\) and \(f(b)\) it must belong to this connected set[5](https://mathworld.wolfram.com/IntermediateValueTheorem.html). In the general form used in the Lean math library, if \(f\) is a continuous function defined on a connected set and takes both values \(\le a\) and values \(\ge a\) on that set, then it equals \(a\) at some point of it[9](https://leanprover-community.github.io/mathlib_docs/topology/algebra/order/intermediate_value.html) • [14](https://people.math.sc.edu/sharpley/math555/Lectures/MetricSpaceConnectedness.html).
## History
The theorem (in the special case \(c = 0\), corresponding to Bolzano's theorem) was first proved by Bernard Bolzano (1781–1848) in 1817[5](https://mathworld.wolfram.com/IntermediateValueTheorem.html). His paper, *Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege*, sought a purely arithmetical proof of the intermediate value theorem, purged of geometric reasoning, and in it Bolzano defined what is now called a [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence); the concept appears in Cauchy's work four years later, and it is unlikely that Cauchy had read Bolzano's work[3](https://mathshistory.st-andrews.ac.uk/Biographies/Bolzano/).
Bolzano did not fully succeed in establishing the theorem. His proof that Cauchy sequences converge to a "constant quantity" was not satisfactory, so a complete proof of the IVT was out of reach; modern historians nonetheless credit the 1817 text with a remarkably advanced understanding of the conditions such a proof requires[10](https://link.springer.com/article/10.1007/s44007-025-00182-w). Cauchy proved the theorem independently, stating it in a more general form in his 1821 *Cours d'analyse*: if \(f(x)\) is continuous between \(x = x_0\) and \(x = X\), and \(b\) is a quantity intermediate between \(f(x_0)\) and \(f(X)\), the equation \(f(x) = b\) can always be satisfied[11](https://fr.wikisource.org/wiki/Cours_d%E2%80%99analyse_de_l%E2%80%99%C3%A9cole_royale_polytechnique/Texte_entier). Neither Bolzano nor Cauchy identified all the assumptions about real numbers that underlay their treatments; that clarification came with the later arithmetization of analysis[4](https://encyclopediaofmath.org/wiki/Arithmetization_of_analysis). Bolzano's techniques were considered especially rigorous for their time but are regarded as nonrigorous by modern standards[5](https://mathworld.wolfram.com/IntermediateValueTheorem.html).
## Applications
**Root finding.** The theorem guarantees that a continuous function with opposite-sign endpoint values has a root in the interval, and the [bisection method](https://www.edgepedia.org/bisection-method) turns this into an algorithm: repeatedly halve the interval, keeping the half on which the sign change persists. If \((p_n)\) is the sequence of midpoints, then \((p_n)\) converges to a zero \(p\) with \(\lvert p - p_n \rvert < (b-a)/2^n\), so each step halves the error bound[2](https://www.geneseo.edu/~aguilar/public/assets/courses/345/numerical-analysis.pdf). The method matters where explicit formulas do not exist; for polynomials of degree five or higher there are in general no formulas for the roots in terms of the coefficients using algebraic operations and radicals[2](https://www.geneseo.edu/~aguilar/public/assets/courses/345/numerical-analysis.pdf).
**Existence results.** A typical classroom application shows that \(f(x) = x - \cos x\) has a zero: \(f(0) = -1 < 0\) and \(f(\pi/2) = \pi/2 > 0\), so a zero lies in \([0, \pi/2]\)[7](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/02%3A_Limits/2.04%3A_Continuity). The same argument shows that every polynomial of odd degree has a real root[8](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/07%3A_Intermediate_and_Extreme_Values/7.02%3A_Proof_of_the_Intermediate_Value_Theorem). Many theorems of calculus are ultimately dependent on the intermediate value theorem[12](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_%28Dumas_and_McCarthy%29/08%3A_New_Page/8.08%3A_New_Page).
**Related properties.** Continuity on an interval implies the Darboux property (the intermediate value property), but the converse fails; for monotone functions, however, the Darboux property on an interval does imply continuity[13](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.09%3A_The_Intermediate_Value_Property).
## Constructive status
In constructive mathematics, where an existence claim must yield a method for finding the object, the classical IVT as stated is not provable: it entails the lesser limited principle of omniscience, an essentially nonconstructive principle[6](https://plato.stanford.edu/entries/mathematics-constructive/). A weak form can be proved constructively by an interval-halving argument, and a stronger constructive version holds for functions that are locally nonzero, meaning that near every point some nearby value of \(f\) differs from zero; this version suffices for most practical purposes[6](https://plato.stanford.edu/entries/mathematics-constructive/). The situation is typical of constructive analysis, where one classical theorem splits into several constructive versions, some or all of which become equivalent under classical logic[6](https://plato.stanford.edu/entries/mathematics-constructive/).
## References
1. [Intermediate Value Theorem — Statement, Proof & Bisection, Unseel](https://unseel.com/mathematics/intermediate-value-theorem)
2. [Numerical Analysis I (Aguilar, SUNY Geneseo), Section 2.1](https://www.geneseo.edu/~aguilar/public/assets/courses/345/numerical-analysis.pdf)
3. [Bernard Bolzano (1781–1848), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Bolzano/)
4. [Arithmetization of analysis, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Arithmetization_of_analysis)
5. [Intermediate Value Theorem, Wolfram MathWorld](https://mathworld.wolfram.com/IntermediateValueTheorem.html)
6. [Constructive Mathematics, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/mathematics-constructive/)
7. [2.4: Continuity, OpenStax Calculus via LibreTexts](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/02%3A_Limits/2.04%3A_Continuity)
8. [7.2: Proof of the Intermediate Value Theorem, Real Analysis (Boman & Rogers) via LibreTexts](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/07%3A_Intermediate_and_Extreme_Values/7.02%3A_Proof_of_the_Intermediate_Value_Theorem)
9. [topology.algebra.order.intermediate_value, mathlib3 documentation](https://leanprover-community.github.io/mathlib_docs/topology/algebra/order/intermediate_value.html)
10. [Bolzano and the Foundation of the Real Continuum, La Matematica (2026)](https://link.springer.com/article/10.1007/s44007-025-00182-w)
11. [Cours d'analyse de l'école royale polytechnique (Cauchy, 1821), Wikisource](https://fr.wikisource.org/wiki/Cours_d%E2%80%99analyse_de_l%E2%80%99%C3%A9cole_royale_polytechnique/Texte_entier)
12. [8.8: Real Functions, Transition to Higher Mathematics via LibreTexts](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_%28Dumas_and_McCarthy%29/08%3A_New_Page/8.08%3A_New_Page)
13. [4.9: The Intermediate Value Property, Mathematical Analysis (Zakon) via LibreTexts](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.09%3A_The_Intermediate_Value_Property)
14. [Connectedness in Metric Space](https://people.math.sc.edu/sharpley/math555/Lectures/MetricSpaceConnectedness.html)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Real analysis*
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