= One-sided representatives of a one-dimensional BV function
{title2=$u_l(t)=c+Du((a,t)),\quad u_r(t)=c+Du((a,t])$}
Both functions represent the same $L^1$ <equivalence class>. <Fubini's theorem> shows that the interval-mass function has <derivative> $Du$, so its difference from $u$ is constant. Continuity of a <finite measure> gives left continuity for $u_l$ and right continuity for $u_r$. Their opposite one-sided limits differ by the <measure atom> $Du(\{t\})$. There are only countably many nonzero <measure atoms>, since every set of <measure atoms> above a fixed magnitude threshold is finite.
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