Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions which result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval.

Note that when f is continuously differentiable (f in C¹([a,b])), this is trivially true by the intermediate value theorem. But even when f′ is not continuous, Darboux's theorem places a severe restriction on what it can be.

Darboux's theorem

Let f : [a,b] → 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 $f,\text{'}$ satisfies the intermediate value property: for every t between $f,\text{'}\_\{+\}(a)$ and $f,\text{'}\_\{-\}(b)$, there is some x in [a,b] such that $f,\text{'}(x)\; =\; t$.

Proof

Without loss of generality we might and shall assume $f,\text{'}\_\{+\}(a)\; >\; t\; >\; f,\text{'}\_\{-\}(b)$. Let g(x) := f(x) - tx. Then $g\text{'}(x)\; =\; f,\text{'}(x)\; -\; t$, $g\text{'}\_\{+\}(a)\; >\; 0\; >\; g\text{'}\_\{-\}(b)$, and we wish to find a zero of $g\text{'}$.

Since g is a continuous function on [a,b], by the extreme value theorem it attains a maximum on [a,b]. This maximum cannot be at a, since $g\text{'}\_\{+\}(a)\; >\; 0$ so g is locally increasing at a. Similarly, , so g is locally decreasing at b and cannot have a maximum at b. So the maximum is attained at some c in (a,b). But then $g\text{'}(c)\; =\; 0$ by Fermat's theorem (stationary points).

See also

• Jean Gaston Darboux
• Intermediate value theorem

External links