Define what it means for a function $f:[0,1] \rightarrow \mathbb{R}$ to be (Riemann) integrable. Prove that $f$ is integrable whenever it is

(a) continuous,

(b) monotonic.

Let $\left\{q_{k}: k \in \mathbb{N}\right\}$ be an enumeration of all rational numbers in $[0,1)$. Define a function $f:[0,1] \rightarrow \mathbb{R}$ by $f(0)=0$,

$$f(x)=\sum_{k \in Q(x)} 2^{-k}, \quad x \in(0,1]$$

where

$$Q(x)=\left\{k \in \mathbb{N}: q_{k} \in[0, x)\right\}$$

Show that $f$ has a point of discontinuity in every interval $I \subset[0,1]$.

Is $f$ integrable? [Justify your answer.]