# One-One function and differentiability

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 $f'(x) \ne 0$ for all $x \in (a, b)$. Then f is 1-1.

True

Discussion:

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

Chinese Remainder Theorem