Inverse function theorem in one dimension

Inverse function theorem (1D): Let $I\subseteq\mathbb{R}$ be an open interval and let $f:I\to\mathbb{R}$ be continuously differentiable. Fix $x_0\in I$ with $f'(x_0)\neq 0$. Then there exist open intervals $J\subseteq I$ and $K\subseteq\mathbb{R}$ with $x_0\in J$ and $f(x_0)\in K$ such that:

1. The restriction $f|_J:J\to K$ is bijective, so it has an inverse function $g:K\to J$.

2. The inverse $g$ is continuously differentiable on $K$, and for all $y\in K$,

   $$g'(y)=\frac{1}{f'(g(y))}.$$

   In particular, $g'(f(x_0))=1/f'(x_0)$.

This result combines local monotonicity from the derivative with the chain rule applied to $f\circ g=\mathrm{id}$. It is the one-dimensional case of the inverse function theorem in higher dimensions .