Question 15
Let $a \lt b$ be real numbers and $f : (a, b) \rightarrow \mathbb{R}$ a function such that the functions $g : (a, b) \rightarrow \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \rightarrow \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing (non-decreasing). Show that the function $f$ is continuous on $(a, b)$.