A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a BV function if there exists $M<\infty$ for which $$\sum_{k=1}^N|f(x_k)-f(x_{k-1})|\leq M$$ for every sequence $x_0<x_1<\ldots<x_N$ and every $N$. Any BV function has only jump discontinuities. That is, at any point $a$, the limit $\lim_{x\rightarrow a^-}f(x)$ and $\lim_{x\rightarrow a^+}f(x)$ exist, but they may or may not be equal.
I want to use this to show that a BV function $f$ is continuous except at countably many points.
Suppose the function is discontinuous at uncountably many points. How can I choose $x_0,x_1,\ldots,x_N$ to contradict the definition of BV?