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.
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 satisfies the intermediate value property: for every t between and , there is some x in [a,b] such that .
Without loss of generality we might and shall assume . Let g(x) := f(x) - tx. Then , , and we wish to find a zero of .
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 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 by Fermat's theorem (stationary points).