Let f be real valued, differentiable on (a, b) and f'(x) \ne 0 for all x \in (a, b) . Then f is 1-1.



Suppose f is not 1-1. Then there exists x_1 , x_2 \in (a, b) such that f(x_1 ) = f(x_2) . Since f(x) is differentiable it must be continuous as well. Applying Rolles Theorem in the interval (x_1 , x_2 ) we conclude that there exists a number c in this interval such that f'(c) = 0. But this contradicts the given conditions. Hence f must be 1-1