Points of discontinuity of a bijective function $f:\mathbb{R} \to [0,\infty)$

We know that the points of discontinuity of a monotone function on an interval $[a,b]$ are countable. Using this can we prov that:

• Any bijection $f: \mathbb{R} \to [0,\infty)$ has infinitely many points of discontinuity.

If yes, how or otherwise how to prove the above result?

Solution 1:

Suppose to the contrary that $f$ has a finite number $n$ of discontinuities, at $x_1, x_2, \ldots, x_n$. Then $\mathbb{R} - \{ x_1, \ldots, x_n \} $ is a union of open intervals $I_1, \ldots I_{n + 1}$. $f$ restricted to each $I_m$ is continuous and injective, and therefore monotone, so each image $f|_{I_m}(I_m)$ is an open (in $\mathbb{R}$) interval $J_m$. The $J_m$ are non-empty, pairwise disjoint and contained in $(0, \infty)$; suppose that $J_{m_1} < J_{m_2} < \ldots < J_{m_{n + 1}}$. Then between each $J_{m_p}$ and $J_{m_{p + 1}}$ there is at least one point, which must be the image of some $x_q$. But this exhausts all the $x$'s, so that $0$ is not in the image of $f$.