Let $f:[a,b]\to\mathbb{R}$ be a real-valued continuous function on $[a,b]$, which is differentiable on $(a,b)$, differentiable from the right at $a$, and differentiable from the left at $b$. Then $f^{{\prime}}$ satisfies the intermediate value theorem: for every $t$ between $f^{{\prime}}_{{+}}(a)$ and $f^{{\prime}}_{{-}}(b)$, there is some $x\in[a,b]$ such that $f^{{\prime}}(x)=t$.

Note that when $f$ is continuously differentiable ($f\in C^{1}([a,b])$), this is trivially true by the intermediate value theorem. But even when $f^{{\prime}}$ is not continuous, Darboux’s theorem places a severe restriction on what it can be.