Continuous Inverse Theorem

Theorem
Let $f$ be a real function defined on some  interval $I$.

Let $f$ be strictly monotone and  continuous on $I$.

Let $g$ be the inverse mapping to $f$.

Then $g$ is strictly monotone and continuous on $J := f \left[{ I }\right]$, where $f \left[{ I }\right]$ denotes the image of $I$ under $f$.

Proof
From Strictly Monotone Function is Bijective, $f$ is a bijection.

From Inverse of Strictly Monotone Function, $f^{-1} : J \to I$ exists and is strictly monotone.

From Surjective Monotone Function is Continuous, $f^{-1}$ is continuous.

Hence the result.