Does there exist a non decreasing function $f:\mathbb{R} \to \mathbb{R}$ such that the set of discontinuities is $\mathbb{Q}$?

Question

Question is to find (if any) $f: \mathbb{R} \to \mathbb{R}$ such that $f$ is non decreasing and the set of discontinuities is $\mathbb{Q}$.

I know a function $f :\mathbb{R} \to \mathbb{R}$ such that it is discontinuous at all rationals and continuous at irrational. But in the above question additional condition is imposed. So I am not able to find one.

Could anyone help me please. Thank you

Answer 1

It is well known to every student of probabilty theory that the function $f(x)=\sum_{\{n:r_n \leq x \}} \frac 1 {2^{n}}$, where $\{r_n\}$ is an ennumertation of $\mathbb Q$ has this property.

Probabilistically this is the distribution function of a random variable that takes the values $r_1,r_2,...$ with probabilites $\frac 1 2, \frac 1 {2^{2}},...$.