Let \(\Omega\subseteq\CC\) be open. A map \(f\colon\Omega\to\CC\) is conformal if \(f\) is holomorphic on \(\Omega\) and \(f'(z)\neq0\) for every \(z\in\Omega\).
A conformal map \(f\colon\Omega\to\CC\) is locally injective: every \(z_0\in\Omega\) has an open neighborhood on which \(f\) is injective.
Since \(f'(z_0)\neq0\), the holomorphic inverse function theorem gives an open neighborhood \(V\) of \(z_0\) such that \(f|_V\) is a bijection onto an open set with holomorphic inverse; in particular \(f|_V\) is injective.
What is a conformal map?