Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a function (not necessarily continuous) from the real line to the real line,
how to prove that the set $V\triangleq \{ a\in \mathbb{R}| \overline{\lim}_{x\mapsto a+} f(x)\neq \overline{\lim}_{x\mapsto a-}f(x) \}$ is countable?