Darboux theorem

If a real-valued function has a finite derivative at each point of an interval on the real axis, then, if the derivative assumes any two values on this interval, this derivative also assumes all the intermediate values on it.

Comments

This is a kind of intermediate value theorem (for the derivative rather than the function). The following theorem holds for functions themselves: If is a continuous mapping between metric spaces, and if is a connected set, then is connected. (See, e.g., [a1].)

References

[a1]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)