If $f^{-1}$ has nowhere zero derivative, then $f$ is differentiable

I think you need to apply theorem 5 from chapter 12 of Spivaks calculus, but not sure how to. If someone could please help me. The question is:

Suppose $f$ is a one-one function and the inverse of $f$ has a derivative which is nowhere zero. Prove that $f$ is differentiable.

Chapter 12, Theorem 5:

Let $f$ be a continuous one-one function defined on an interval, and suppose that $f$ is differentiable at $f^{-1}(b)$, with derivative $f'(f^{-1}(b))\neq 0$. Then $f^{-1}$ is differentiable at $b$, and $$(f^{-1})'(b) = \frac{1}{f'(f^{-1}(b))}.$$

We make the following observations:

1. If $f$ is one-to-one then $f^{-1}$ is one-to-one.

2. If $f^{-1}$ is one-to-one then $f^{-1}$ has an inverse, and $(f^{-1})^{-1}=f$.

Now this is a simple case of applying Theorem 5 to the function $f^{-1}$.

• To apply Theorem 5 directly you need continuity of $f^{-1}$, and this follows from differentiability. Jan 16, 2013 at 5:29
• @JonasMeyer I thought about making that remark, then figured that the OP should do at least that much :) Jan 16, 2013 at 5:30
• Thank you for the answer and editing my question.
– Dick
Jan 16, 2013 at 5:33
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