Can the inverse of a differentiable function be non continuous?

Question

I have two questions:

1. Suppose that $f:(a,b)\rightarrow A$, $(A\subseteq\mathbb{R})$ is a differentiable real function and suppose that $f$ is invertible. Can the inverse $f^{-1}$ be non-continuous?
2. Is there a difference between the Inverse function theorem for real variable and the Inverse function theorem for complex variable? Is there an extra requirement for the theorem to be true? Some statements regarding complex variable come with less restrictions. Is the Inverse function theorem is one of them?

Thanks!

Answer 1

• No. Differentiability isn't even needed. If $f$ is continuous and montone on $(a,b)$, then $f$ is an open map (hint: prove that $f$ maps open intervals to open intervals).

• In a sense, the assumptions and conclusions are the same in both theorems, because a holomorphic function is the complex-analytic equivalent of a real differentiable function. The complex-analytic version certainly has consequences of which there isn't a counterpart in real analysis, but those consequences come from the well-known peculiar results in complex analysis (holomorphy = infinite differentiability, Rouche's theorem, etc.)