real analysis - Can we construct a function $f:mathbb{R} rightarrow mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

I think it is probable because we can consider

$$y = \begin{cases} \sin \left( \frac{1}{x} \right), & \text{if } x \neq 0, \\ 0, & \text{if } x=0. \end{cases}$$

This function has intermediate value property but is discontinuous on $x=0$.

Inspired by this example, let $r_n$ denote the rational number,and define

$$y = \begin{cases} \sum_{n=1}^{\infty} \frac{1}{2^n} \left| \sin \left( \frac{1}{x-r_n} \right) \right|, & \text{if } x \notin \mathbb{Q}, \\ 0, & \mbox{if }x \in \mathbb{Q}. \end{cases}$$

It is easy to see this function is discontinuons if $x$ is not a rational number. But I can't verify its intermediate value property.