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 $f'$.

Proof sketch: Define $h(x)=f(x)-\alpha x$ on $[x_1,x_2]$. Then $h$ is continuous on $[x_1,x_2]$ and differentiable on $(x_1,x_2)$, with $h'(x_1)=f'(x_1)-\alpha$ and $h'(x_2)=f'(x_2)-\alpha$ of opposite signs. The function $h$ attains a minimum on $[x_1,x_2]$ (by the extreme value theorem ); that minimum occurs at some $c\in(x_1,x_2)$, and differentiability forces $h'(c)=0$, i.e. $f'(c)=\alpha$.