Exercise 1954Calculus › Single-variable calculus › Differentiation › Mean Value TheoremM1Added 4 Nov 2026

Darboux's theorem on derivatives

#Darboux#intermediate value#derivatives

Statement

Show that the derivative $f'$ of a differentiable function takes every intermediate value (Darboux's theorem), even if $f'$ is not continuous.

Hint

Let $a<b$ and $f'(a)<k<f'(b)$ . Consider $g(x)=f(x)-kx$ ; show $g'$ vanishes somewhere in $]a,b[$ .

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Solution

Setup. WLOG $f'(a)<k<f'(b)$ . Let $g(x)=f(x)-kx$ ; then $g'(a)<0<g'(b)$ . The function $g$ decreases near $a$ (right) and increases near $b$ (left), so $g$ attains a minimum at some $c\in]a,b[$ .

Derivative at minimum. $g'(c)=0$ , so $f'(c)=k$ .

Conclusion. $f'$ attains the intermediate value $k$ .

Remark. Hence $f'$ has the intermediate value property even when it is not continuous (e.g. $x^2\sin(1/x)$ near the origin).

Darboux's theorem on derivatives

Statement

Solution

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