Discontinuity set of an increasing right continuous function

Question

Let $f:[0,\infty) \rightarrow [0,\infty] $ be both right-continuous and increasing. Is it possible that $f$ is discontinuous at dense subset of $\mathbb R$? I already know that if we drop right-continuity, then $f$ can be discontinuous at every rational.

Any hint would be appreciated. Thanks and regards.

Answer 1

This should work:

$f(x)=\sum_{q_n \leq x} 2^{-n},$ where $\{q_n\}$ is an enumeration of the rationals $\mathbb{Q} \cap [0,+\infty)$.

Tell me if you have some doubts.