Intermediate value theorem
Suppose is a continuous function and a closed interval is contained in the domain of (in particular, the restriction of to the interval is continuous). Then, for any between the values and (see note below), there exists such that .
Note: When we say is between and , we mean if and we mean that if .
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 where is the signum function and we define it to be zero at 0.
Here, we consider the domain , with and , but there is no satisfying , even though .
The function needs to be defined throughout the domain
Consider the function . We have and . However, there is no such that . The reason the theorem fails is that is not defined at the point 0, and hence it is not defined on the domain .