Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if ${\displaystyle f}$ is a continuous function whose domain contains the interval [a, b], then it takes on any given value between ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$ at some point within the interval.

This has two important corollaries:

1. If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).
2. The image of a continuous function over an interval is itself an interval.