Inverse function theorem in one dimension

Inverse function theorem (one dimension). Let $I\subseteq\mathbb R$ be an open interval and let $f:I\to\mathbb R$ be continuously differentiable. If $x_0\in I$ and $f'(x_0)\ne0$, then there are open intervals $J\subseteq I$ and $K\subseteq\mathbb R$, containing $x_0$ and $f(x_0)$, respectively, such that:

1. The restriction $f|_J:J\to K$ is bijective.
2. Its inverse $g:K\to J$ is continuously differentiable and satisfies
   $$g'(y)=\frac{1}{f'(g(y))}.$$