The example of the bijective mapping $f: \mathbb{R}\to\mathbb{R}$ such that $f$ is everywhere discontinuous, but an inverse of the $f$ is continuous at a countable set of points

In this paper we consider the example of the bijective mapping $f: \mathbb{R}\to\mathbb{R}$ such that $f$ is everywhere discontinuous, but an inverse of the $f$ is continuous at a countable set of points.