Sequence formed by points of continuity

Question

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be a monotone increasing function. It is well known that $f$ has at most a countably infinite number of discontinuities. I would like to know if, given a discontinuity point $a$, there are sequences formed by points of continuity $(a_n)$ and $(b_n)$ such that $a_n<a$, $a<b_n$ and $a_n \rightarrow a$, $b_n \rightarrow a$ as $n \rightarrow \infty$.

Answer 1

Given the interval $\left(a-\frac{1}{n},a\right),$ there are uncountably many points, and only countably many points where $f$ is discontinuous, so we can find an $a_n\in\left(a-\frac{1}{n},a\right)$ such that $f$ is continuous at $a_n.$ Then $a_n<a$ for all $n$ and $a_n\to a.$

Similarly, choose $b_n\in\left(a,a+\frac{1}{n}\right)$ where $f$ is continuous, and then $b_n>a$ and $b_n\to a.$

Answer 2

If the function has only finitely many discontinuities in total, then the only such sequences are those that eventually (i.e., starting with some index) constantly equal $a$.