# Equivalent definitions of limit and Discontinuities of monotonic functions

I have two questions.

1. PMA, Rudin p.97 proves the existence of a monotonic real function $f$ defined on $(a,b)$ such that $f$ is continuous on $E$- a countable subset of $(a,b)$ and discontinuous on $(a,b) \setminus E$.

Then, it states that "It should be noted that the discontinuities of a monotonic function need not be isolated".