In mathematical analysis, the intermediate value theorem states that if ${\displaystyle f}$ is a continuous function whose domain contains the interval [a, b] and ${\displaystyle s}$ is a number such that ${\displaystyle f(a)<s<f(b)}$, then there exists some ${\displaystyle x}$ between ${\displaystyle a}$ and ${\displaystyle b}$ such that ${\displaystyle f(x)=s}$. That is, the image of a continuous function over an interval is itself an interval that contains ${\displaystyle f(a),f(b)}$.

For example, suppose that ${\displaystyle f\in C([1,2]),f(1)=3,f(2)=5}$ , then the graph of ${\displaystyle y=f(x)}$ must pass through the horizontal line ${\displaystyle y=4}$ while ${\displaystyle x}$ moves from ${\displaystyle 1}$ to ${\displaystyle 2}$. Over the interval, the set of function values has no gap, and the graph can be drawn without lifting a pencil from the paper.

The corollary Bolzano's theorem states that if a continuous function has values of opposite sign inside an interval, then it has a root in that interval. The theorem depends on, and is equivalent to, the completeness of the real numbers, although Weierstrass Nullstellensatz is a version of the intermediate value theorem for polynomials over a real closed field.

A similar result to the intermediate value theorem is the Borsuk–Ulam theorem, which underpins why rotating a wobbly table will always bring it to stability. Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property, even though they need not be continuous.

This captures an intuitive property of continuous functions over the real numbers: given ${\displaystyle f}$ continuous on ${\displaystyle [1,2]}$ with the known values ${\displaystyle f(1)=3}$ and ${\displaystyle f(2)=5}$, then the graph of ${\displaystyle y=f(x)}$ must pass through the horizontal line ${\displaystyle y=4}$ while ${\displaystyle x}$ moves from ${\displaystyle 1}$ to ${\displaystyle 2}$. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.^{[citation needed]}

The intermediate value theorem states the following:^{[citation needed]}

Consider the closed interval ${\displaystyle I=[a,b]}$ of real numbers ${\displaystyle \mathbb {R} }$ and a continuous function ${\displaystyle f\colon I\to \mathbb {R} }$. Then

Version I is naturally contained in Version II.