Darboux's Theorem

Darboux’s Theorem: Let $f:(a,b)\to\mathbb{R}$ be differentiable . If $x_1,x_2\in(a,b)$ with $x_1<x_2$ and $\alpha$ lies between $f'(x_1)$ and $f'(x_2)$, then there exists $c\in(x_1,x_2)$ such that $f'(c)=\alpha.$

This theorem says derivatives cannot have jump discontinuities: they may be very irregular, but they still take all intermediate values . It is a key qualitative property of differentiation that does not rely on continuity of $f'$.