Single variable $C^1$ locally invertible function is globally invertibe? I am wondering if and why a single variable $C^1$ locally invertible function on the entire real line is globally invertible.
I've been told that for single variable function, if the derivative is always non zero and continuous, then the inverse can be defined on the entire range of
the function.
I know quite well that local invertiblity does not imply global invertibility in general.

That's correct for $f:\mathbb{R}\rightarrow \mathbb{R}$. If $f^\prime$ is everywhere nonzero $f$ is strictly increasing or decreasing. It's easy to see that this implies that $f$ is one to one (injective), hence a unique inverse  $f(\mathbb{R}) \rightarrow \mathbb{R}$ exists (this is just set theory). That this is continuously differentiable is only a local question.
(Of course you cannot conclude that $f$ is onto, which can be seen from simple examples)
