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Derivatives satisfy the intermediate value property even when they are not continuous.
Darboux's theorem. Let be an interval, and let be differentiable. If lie in the interior of and lies between and , then there is 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.
Intermediate value theorem: Let be continuous on . If is any number between and (that is, ), then there exists such that
A continuous map between topological spaces and is a function such that for every open set , the preimage is open in .
Let , let , and let . The point is a discontinuity point of if is not continuous at . Equivalently, there exists such that for every there is with