Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the binary system, then set $$f(x)=\lim_{n\to\infty}\frac{x_1+\cdots+x_n}{n}$$ if the limit exists and belongs to $(0,1)$, and set $f(x)$ to (say) $\frac{1}{2}$ otherwise.

Is this example known (I suppose it is), and what's the reference for it?