One-sided representatives of a one-dimensional BV function ($u_l(t)=c+Du((a,t)),u_r(t)=c+Du((a,t])$)

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.