Inverse function theorem

Let \(I\subseteq\RR\) be an open interval, let \(f\in C^1(I)\), and let \(a\in I\) with \(f'(a)\neq0\). Then there are open intervals \(U\ni a\) and \(V\ni b\coloneqq f(a)\) such that \(f\) restricts to a bijection \(U\to V\) whose inverse \(g\colon V\to U\) is \(C^1\), and \[ g'(b)=\frac{1}{f'(a)}. \]

Let \(A\subseteq\RR^n\) be open, let \(F\colon A\to\RR^n\) be \(C^1\), and let \(a\in A\) with \(\det DF(a)\neq0\). Then there are open sets \(U\ni a\) and \(V\ni F(a)\) such that \(F\) restricts to a bijection \(U\to V\) whose inverse \(F^{-1}\colon V\to U\) is \(C^1\), and for every \(p\in U\), \[ D(F^{-1})(F(p))=\bigl(DF(p)\bigr)^{-1}. \]

Let \(\Omega\subseteq\CC\) be open, let \(f\) be holomorphic on \(\Omega\), and let \(p\in\Omega\) with \(f'(p)\neq0\). Then there is an open set \(V\ni p\) such that \(f(V)\) is open and \(f|_V\colon V\to f(V)\) is a biholomorphism.