Inspired By Pi Han Goh

$f(x)=\begin{cases} x-2 & \textrm{if } x < 1 \\ x & \textrm{if } x \geq 1 \\ \end{cases}$ is a discontinuous function which is increasing (in fact, strictly increasing).

But an increasing function is only discontinuous at a countable number of points at most....it is continuous almost everywhere, in a Lebesgue measure sense.