Intermediate value theorem

Statement

Full version

Suppose $f$ is a continuous function and a closed interval $[a,b]$ is contained in the domain of $f$ (in particular, the restriction of $f$ to the interval $[a,b]$ is continuous). Then, for any $t$ between the values $f(a)$ and $f(b)$ (see note below), there exists $c \in [a,b]$ such that $f(c) = t$.

Note: When we say $t$ is between $f(a)$ and $f(b)$, we mean $t \in [f(a),f(b)]$ if $f(a) \le f(b)$ and we mean that $t \in [f(b),f(a)]$ if $f(b) \le f(a)$.

Short version

Any continuous function on an interval satisfies the intermediate value property.

Caveats

The statement need not be true for a discontinuous function

It is possible for a function having a discontinuity to violate the intermediate value theorem. Below is an example, of the function $f(x) := (x + 1)\operatorname{sgn}(x)$ where $\operatorname{sgn}$ is the signum function and we define it to be zero at 0.

Here, we consider the domain $[a,b] = [-1,3]$, with $f(-1) = 0$ and $f(3) = 4$, but there is no $c \in [-1,3]$ satisfying $f(c) = 1/2$, even though $1/2 \in [0,4]$.

The function needs to be defined throughout the domain

Consider the function $f(x) := 1/x$. We have $f(-1) = -1$ and $f(1) = 1$. However, there is no $c \in [-1,1]$ such that $f(c) = 1/2$. The reason the theorem fails is that $f$ is not defined at the point 0, and hence it is not defined on the domain $[-1,1]$.