Given the following:
If $f : (a,b)\rightarrow \mathbb{R}$ is monotone, then $f$ has at most countably many points of discontinuity in (a,b), all of which are jump discontinuities.
We must deduce that a monotone $f: \mathbb{R} \rightarrow \mathbb{R}$ has points of continuity in every open interval.
My work:
W.L.O.G. assume $f$ is increasing. Because $f$ is monotone (in this case increasing), I know that the only discontinuities that can exist are jump discontinuities (meaning that the left hand limit is strictly less than the right hand limit). We also know that these limits exist. The intervals of discontinuity are also disjoint. Given an interval $(a,b)$, a point $c \in (a,b)$ can be discontinuous if it meets the criteria mentioned above; that is $f(c-) < f(c+)$. There are points in an interval $(x,c)$ and $(c,y)$ which are continuous (for the limits to exist); where $x,y \in (a,b)$
I'm sure I'm missing a lot of fine details, and may even have the wrong idea on how to approach it. Any critique and clarification would be greatly appreciated.
Best Answer
HINT: You know that $f$ has at most countably many discontinuities in $(a,b)$. Is $(a,b)$ countable?