Darboux's theorem (analysis)
Theorem in real analysis
Nº Q660799 ★
Common · Knowledge
Darboux's theorem (analysis)
Theorem in real analysis
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 {\displaystyle f} is continuously differentiable, this is a consequence of the intermediate value theorem.
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From Wikipedia
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 {\displaystyle f} is continuously differentiable, this is a consequence of the intermediate value theorem. But even when f ′ {\displaystyle f'} is not continuous, Darboux's theorem places a restriction on the behaviour of f ′ {\displaystyle f'} over any closed interval.
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