Darboux's theorem (analysis)

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This article is about Darboux's theorem in real analysis. For Darboux's theorem in symplectic topology, see Darboux's theorem.

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.

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

[edit] Proof

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

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' + (a) > 0 so g is locally increasing at a. Similarly, g' − (b) < 0, 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'(c) = 0 by Fermat's theorem (stationary points).

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