Inverse of a Continuous Strictly Increasing Function on an Interval: Continuity and Derivative

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $I$ be an interval containing two distinct points, let $f:I\to\mathbb{R}$ be continuous on $I$ and strictly increasing, that is, $f(s)<f(t)$ for all $s,t\in I$ with $s<t$, and let $J=f(I)$ be the image of $I$ under $f$.

(i) $J$ is an interval containing two distinct points.

(ii) $f$, read as a map $I\to J$, is bijective.

(iii) The inverse $g=f^{-1}:J\to I$ of this bijection is strictly increasing: $g(y)<g(z)$ for all $y,z\in J$ with $y<z$.

(iv) The map $g$, read as a map $J\to\mathbb{R}$, is continuous on $J$.

(v) Let $x\in I$. If $f$ is differentiable at $x$ and $f'(x)\neq0$, then $g$, read as a map $J\to\mathbb{R}$, is differentiable at $f(x)$, and $g'(f(x))=1/f'(x)$.