Darboux’s theorem (analysis)

Let $f:[a,b]\to ℝ$ 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 $f^{\prime }$ 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.