Darboux's theorem (analysis)

This article is about Darboux's theorem related to the intermediate value theorem. For Darboux's theorem in differential geometry, see Darboux's theorem.

In real analysis, Darboux's theorem states that the derivative of any real-valued function of a real variable has the intermediate value property, that is, that the image of an interval is also an interval.

When is continuously differentiable, this is a consequence of the intermediate value theorem. But even when is not continuous, Darboux's theorem places a restriction on the behaviour of over any closed interval.

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