Suppose $f$ is defined, continuous, and strictly monotone increasing on $(a,b)$ such that $$ c = \lim\limits_{x\to a^+} f(x) \text{ and } d = \lim\limits_{x\to b^-} f(x) $$ exist in the extended reals. Prove the domain of $f^-1$ is $(c,d)$ and $$ \lim\limits_{x\to c^+} f^{-1}(x) = a \text{ and } \lim\limits_{x\to d^-} f^{-1}(x) = b. $$
Given $f$ is strictly monotone increasing on $(a,b)$, $f^{-1}$ is strictly monotone increasing on the range of this interval. Can you claim that the range of $f$ is $(c,d)$ given the first set of limits? If so, why? I believe we also need the fact that $f$ is one-to-one to show that $f^{-1}$ is one-to-one, but I am not sure where to go from there.
A strictly increasing function is necessarily one-to-one and its inverse is also strictly increasing and one-to-one.