In this paper http://www.ams.org/journals/jams/1998-11-03/S0894-0347-98-00269-0/S0894-0347-98-00269-0.pdf, S. Solocki define Lebesgue's functions as follows":
Let $Q$ be the set of all points in $2^{\omega}$ which are eventually equal to 1. For each $x\in Q$ fix a number $a_x>0$ so that
- if $x, y \in Q$ , $x\neq y$ then $a_x \neq a_y$,
- $a_x <\frac{1}{3^{n_0}}$, where $n_0$ is the smallest natural number such that $x(n)=1$ for $n \geq n_0$.
Let $H: 2^{\omega} \rightarrow [0,1]$ be the well-known embedding $H(X)= \sum_{n=0}^{\infty} \frac{x(n)}{3^{n+1}}$. Let $L, L_1: 2^{\omega} \rightarrow \mathbb{R}$ by letting: $$L(x)=\begin{cases} H(x), &\mbox{ if } x\not\in Q\\ H(x)+ a_x,& \mbox{ if }x\in Q. \end{cases}$$
and $$L_1(x)=\begin{cases} 0, &\mbox{ if } x\not\in Q\\ a_x,& \mbox{ if }x\in Q. \end{cases}$$
My question: are $L, L_1$ Baire one function?
I think yes. Because the form of functions $L, L_1$ look like the form of the Riemann function but I cannot prove it.
Anybody know how to prove it? I think that it is easier if we prove it in $\varepsilon-\delta$ terms instead of show that those functions are limit of a sequence of continuous functions.
The following definition of Baire class one in $\varepsilon-\delta$ terms: $f: X \rightarrow Y$ from a topological space $X$ into a metric space $(Y,d)$, is Baire class one if for any $\varepsilon >0$, there exists a positive function $\delta$ on $X$ such that for any $x,y \in X$, $$d(f(x),f(y)) <\varepsilon, \mbox{ whenever } |x-y|< \delta(y)\wedge \delta(x).$$