Suppose that $f : \mathbb{R} → \mathbb{R}$ is a continuous function that is strictly increasing (i.e. $f(x) < f(y)$ whenever $x < y$). Find a necessary and sufficient condition for $f$ to have a strictly increasing continuous inverse function $g : \mathbb{R} → \mathbb{R}$.
This is a question which appeared in my practice test, and I am not sure why we need a condition as $f$ seems sufficiently strong as stated. Is there a stronger condition needed?