Find function $f:[0,1]\to \mathbb R$, whose set of discontinuities is such that $\left\{ \frac{k}{2^n}; \quad k,n \in \mathbb N, k \le 2^n \right\}$

Question

Find function $f\colon[0,1]\to \mathbb R$, whose set of discontinuities is such that

$$\left\{\frac{k}{2^n}; \quad k,n \in \mathbb N, k \le 2^n\right \}$$

I guess that it is Riemann function or something like that but I have no clue how it would look like

Answer 1

Let $f(x) = 0$ whenever $x$ is not a dyadic fraction, and $f\left(\frac{k}{2^n}\right) = 2^{-n}$.