ある本に，「Since a differentiable function on an interval in R with nowhere zero derivative has a differentiable inverse, it is tempting to think
that if the derivative f'(a) ≠ 0, then f should have a local inverse at a.」と書いてあるのですが，「a differentiable function on an interval in R with
nowhere zero derivative has a differentiable inverse」はどうやって証明するのでしょうか？