Yeh in his book Real Analysis defines a right-continuous step function $f$ on a finite interval $[a, b) $ as of the form $f= \sum_{k=1}^n c_k\cdot \mathbf 1 _{I_k}, $ where $\sqcup_{k=1}^n I_k=[a, b) $ and $I_k$s are all closed-open intervals like $[a, b). $ Any function on $\mathbb R$ is right-continuous step function if its restriction on any finite subinterval of the form $[a, b) $ is right continuous step function.
The author then notes
[T]he set of points of discontinuity of a right-continuous step function is a countable set having no limit points in $\mathbb R. $
Now in proving a result due to Borel showing a lebesgue measurable function being approximated by a continuous function, the author again brings the above statement:
I didn't understand the necessity of the set of end points of the intervals $I_n$ not having limit points: why should it matter? Couldn't the intervals be numbered such that $I_n$ precedes $I_{n+1}$ without worrying about whether the end points have limit points or not? Feels trivial to me: just re index the intervals if needed.
Why does the author present the argument based on the absence of limit points for numbering the intervals?