I have a sequence $(f_n)_n : [0,1] \to [0,1]$ of strictly increasing functions.
$f_n$s are, in general, not left or right continuous. If they were right continuous, from what I know, we have results that show that there exists a function $f$ such that $f_n \to f$ in the weak* topology.
Are there any results that suggest that this sequence $f_n$ converges to $f$ in some sense? Would it help if I knew that $f_n$s are usc?