The problem is the following:
If $f$ is a real-valued function of bounded variation on an interval $[a,b] \subset \mathbb{R} $. Show that the set $$ A=\{y \in \mathbb{R} : \{x \in [a, b] : f(x)=y \}\: \text{is an infinite set} \} $$ has Lebesgue measure zero.
During my attempts, I felt that I was not familiar enough with functions of bounded variation and how to argue that one set has Lebesgue measure zero or not.
Any help is appreciated!
