Darboux's theorem (analysis)

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

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