Increasing functions are Baire one $\newcommand{\R}{\mathbb R}$ 
Let $f: \R \to \R$ be an increasing function. $(a \leq b \Rightarrow f(a) \leq f(b))$. I want to prove that $f \in \mathcal B$, the Baire one functions.
I first tried to prove that $g(x) := \lfloor f(x)+x \rfloor$ is in $\mathcal B$. The points of discontinuity of $g$ are countable since $g$ is increasing. Now, let $\mathcal D$ be the points in $\R$ where $g$ is not continuous. If $z \in \mathcal D$ then I want to prove that $\exists \delta > 0: (z-\delta,z+\delta) \cap \mathcal D = \{z\}$, i.e. $z$ is the only discontinuity in a certain neighborhood. Then it is not hard to prove that $g \in \mathcal B$ and that $f$ is the uniform limit of $\mathcal B$ functions and thus is also $\mathcal B$.
I already tried to prove this (existence of $\delta > 0$) by contradiction. However, I could not see a nice way to get here a contradiction.
 A: I explained in a comment why your approach cannot work; here’s a start on one that does.
Let $D_0$ be the set of points at which the monotonically increasing function $f$ is discontinuous, let $D_1$ be a countable dense subset of $\Bbb R\setminus D_0$, and let $D=D_0\cup D_1$. Note that we may assume that $n,-n\in D$ for each $n\in\Bbb Z^+$. (Why?) $D$ is countably infinite, so we can enumerate it as $D=\{x_n:n\in\Bbb N\}$. For $n\in\Bbb N$ let $D_n=\{x_k:k\le n\}$.
For each $n\in\Bbb Z^+$ there is an $m_n\in\Bbb N$ such that $n,-n\in D_{m_n}$ and for each $x\in[-n,n]$ there is an $x_k\in D_{n_m}$ such that $|x-x_k|<\frac1{2^n}$. (Why?) We may further assume that $m_{n+1}>m_n$ for all $n\in\Bbb Z^+$. Let $E_n=D_{m_n}\cap[-n,n]$; we can enumerate $E_n=\{y_k^{(n)}:0\le k\le e_n\}$, where $e_n=|E_n|-1$, in such a way that $$-n=y_0^{(n)}<y_1^{(n)}<\ldots<y_{e_n-1}^{(n)}<y_{e_n}^{(n)}=n\;.$$ 
Let $f_n:\Bbb R\to\Bbb R$ be defined as follows:


*

*$f_n(x)=f(-n)$ if $x\le -n$;  

*$f_n(x)=f(n)$ if $x\ge n$;  

*$f_n\left(y_k^{(n)}\right)=f\left(y_k^{(n)}\right)$ for $k=0,\ldots,e_n$; and  

*$f_n$ is linear on each interval $\left[y_k^{(n)},y_{k+1}^{(n)}\right]$ for $k=0,\ldots,e_n-1$.


Clearly $f_n$ is continuous. Show that $\langle f_n:n\in\Bbb Z^+\rangle$ converges pointwise to $f$.
