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A continuous function on an interval takes all values between its endpoint values.
Intermediate value theorem: Let be continuous on . If is any number between and (that is, ), then there exists such that
This is one of the basic consequences of being a continuous map on an interval. A notable application is Darboux's theorem, which shows that derivatives also satisfy an intermediate value property.
A continuous map between topological spaces and is a function such that for every open set , the preimage is open in .
Darboux's theorem: Let be an interval, and let be differentiable on . Then the derivative has the intermediate value property: whenever are in and lies between and , there exists such that
Thus behaves like a function satisfying the intermediate value theorem, even though need not be a continuous map and may have points of discontinuity.