I've found this problem while I was reading a paragraph about Riemann integration on some notes a mate gave me a long time ago.
Let $f \colon [a,b] \to \mathbb R$ be a bounded function. Suppose there exists a sequence of step functions $f_k$ s.t. $ f(x)=\sup_{k \in \mathbb N} f_k(x) \quad \forall x \in [a,b] $ Show that $f$ is lower semicontinuous in $[a,b]\setminus C$ where $C$ is a at most countable subset.
The problem seems quite easy: indeed, the pointwise sup of a family of lower semicontinuous functions is still semicontinuous (it's just a consequence of topology axioms, intersection of closed sets is closed). In other words, if $x$ is a point in which $f_k$ is l.s.c. for every $k \in \mathbb N$ so is $f$.
But what about the set $C$? Initially, I thought that the points of $C$ were the points of "jump" for $f_k$ (for some $k \in \mathbb N$). But is it true that a step function is not lower semicontinuous in a "jump" point? Is the problem clear? Hope so.
Thanks in advance.