Discontinuities of an injective function from $\mathbb{R}$ to $\mathbb{R}$

Question

It is well known that a monotonic function from $\mathbb{R}$ to $\mathbb{R}$ can have only countably many discontinuities.

Question: Is it true that an injective function from $\mathbb{R}$ to $\mathbb{R}$ can have countable many discontinuities?

Answer 1

A counterexample was given by David Mitra: $$f(x)=\begin{cases}x,\quad & x\in\mathbb{Q} \\ x+1 , \quad &x\notin\mathbb{Q}\end{cases}$$ This is an everywhere discontinuous bijection, the inverse being $$f^{-1}(x)=\begin{cases}x,\quad & x\in\mathbb{Q} \\ x-1 , \quad &x\notin\mathbb{Q}\end{cases}$$