TITLE: Is there a function defined on real numbers which is continuous from the left, but not from the right, everywhere
QUESTION [23 upvotes]: I am teaching Mathematical analysis. A student asked this question. I think this is a good question, but don't know the answer.

REPLY [14 votes]: Let $\omega(f,A):= \sup \{|f(x)-f(y)| : x,y \in A\}$ be the oscillation of $f$ on $A$, and let $O_n$ be the union of all open sets $A$ s.t. $\omega(f,A) < 1/n$. Then $O_n$ is open and its complement is countable because every uncountable closed set contains a point that is a limit point from the left of the set, while for every real $x$ there is a $y<x$ s.t. $(y,x)$ is a subset of $O_n$. The function $f$ is continuous at points of $B:=\cap O_n$, and the complemented of $B$ is countable.