I am reading "Calculus 4th Edition" by Michael Spivak.
The following theorem is on p.237 in this book:
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 $$(f^{-1})^{'}(b)=\frac{1}{f^{'}(f^{-1}(b))}.$$
When I read the statement of THEOREM 5, I wondered why many calculus books don't contain the following THEOREM.
Why?
THEOREM
Let $f$ be a continuous one-one function defined on an interval, and suppose that $\lim_{h\to 0}\frac{f(f^{-1}(b)+h)-f(f^{-1}(b))}{h}=\pm\infty.$ Then $f^{-1}$ is differentiable at $b,$ and $$(f^{-1})^{'}(b)=0.$$
