Darboux's Theorem states that a function that is the derivative of some differentiable function on an interval must satisfy the intermediate value property, even where the derivative itself is not continuous. Named for Jean Gaston Darboux, it shows that derivatives, while not always continuous functions, can never have a jump discontinuity.
Facts
StatementThe derivative of a differentiable real valued function has the intermediate value property, so it takes every value between the derivative's values at the endpoints of any interval, even where the derivative itself is not continuous. 1 Classification
Statement Form Connections
Sources
1. Darboux's Theorem, Analysis (Wikipedia)
Wikimedia FoundationStatement of the theorem section
The original proof by Jean Gaston Darboux was published in 1875.
Lead section
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.
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