Darboux's theorem (analysis)

In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.

When ${\displaystyle f}$ is continuously differentiable (i.e. ${\displaystyle f\in C^{1}[a,b]}$), this is a consequence of the intermediate value theorem. But even when ${\displaystyle f'}$ is not continuous, Darboux's theorem places a severe restriction on what it can be.