Let's understand one-one function and differentiability with the help of a problem. Try it yourself before reading the solution.
Let f be real valued, differentiable on (a, b) and for all . Then f is 1-1.
Suppose f is not 1-1. Then there exists such that . Since f(x) is differentiable it must be continuous as well. Applying Rolles Theorem in the interval 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