baire $1$ function Here is a new definition of Baire Class one function. Suppose that $X$ is a complete separable metric space. A function $f:X \rightarrow \mathbb{R}$  is said to be Baire class one if for any $\epsilon>0$, there exists $\delta:X \rightarrow \mathbb{R}$ such that for any $x,y \in X$, 
$$d(x,y)<\delta(x) \wedge \delta(y) \Rightarrow |f(x)-f(y)|<\epsilon$$
Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ given by 
\begin{cases} 
      0 & 0 \leq x < 1 \\
      1 & x=1 \\
   \end{cases}
Clearly $f$ is a Baire one function, as it is the pointwise limit of the sequence of continuous functions $f_n(x)=x^n$. However, I fail to verify that it is a Baire one function using the definition given above, that is, unable to find $\delta:\mathbb{R} \rightarrow \mathbb{R}$. Can anyone help?
 A: Rather old and easy question to remain unanswered for this long.  I was searching for something else and this popped up.  May as well finish it off. I expect the poster  Mr Idonknow is, after all this time, probably Mr Iamuchbetterinformed and not so interested anymore.
Consider the function (similar to the one requested):
$$f(x) =  \begin{cases} 
      0 &  x < 0\\
      1 & x\geq 0 \\
   \end{cases}$$
This is clearly Baire 1.   Easy to find a sequence of continuous functions converging pointwise to it.   Or it is easy to investigate the sets
$\{x:  f(x)<\alpha\}$.
Let us check how to use this equivalent definition.
Let $\epsilon>0$.    For each  $x\not= 0 $  choose  $ \delta(x)=|x|/2$  and then choose   $\delta(0)=1$.    Note that if
$$d(x,y)< \min\{\delta(x),  \delta(y) \}$$
then either both  $x$  and $y$ are negative or both $x$ and $y$ are positive.  It follows that   $|f(x)-f(y)| = 0 < \epsilon$.  From that criterion it follows that the function is Baire 1.
So why did I hesitate on this trivial little problem?    This particular expression using a guage $\delta$ in relation to Baire one functions (and other properties) occurs repeatedly in this paper:
A. M. Bruckner, R. J. O'Malley and B. S. Thomson. Path Derivatives: A Unified View of Certain Generalized Derivatives.
Transactions of the American Mathematical Society, Vol. 283, No. 1 (May, 1984),  97-125.
In fact Theorem 5.2 of that paper could also be used to show this function is Baire 1 (but that would be overkill).
