Proposition 6

Let $f$ be a homeomorphism of an interval $I \subset \mathbf{R}$ onto an interval $J = f(I) \subset \mathbf{R}$, and let $g$ be the inverse homeomorphism$^3$. If $f$ is differentiable at the point $x_0 \in I$, and if $f'(x_0) \neq 0$, then $g$ has a derivative equal to $1/f'(x_0)$ at $y_0 = f(x_0)$.

For each $y \in J$ we have $g(y) \in I$ and $u = f(g(y))$; we thus can write

for $y \neq y_0$. When $y$ tends to $y_0$ while remaining in $J$ and $\neq y_0$, then $g(y)$ tends to $x_0$ remaining in $I$ and $\neq x_0$, and the right-hand side in the preceding formula thus has limit $1/f'(x_0)$, since by hypothesis $f'(x_0) \neq 0$.