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Put $\mu =Du$, a finite signed Radon measure, and defineFor $\phi \in C_c^1((a,b))$, Fubini's theorem for the signed measure givesThus $D{F}_l=\mu =Du$. A distribution on a connected interval with zero derivative is constant: any compactly supported test of integral zero is the derivative of a compactly supported test, so it pairs to zero with $u-F_l$. Choose the resulting constant $c$. Thenare one-sided representatives of a one-dimensional BV function. The first equals $u$ almost everywhere. Their difference is $\mu (\{t\})$, nonzero at at most countably many points, so the second also equals $u$ almost everywhere.

Finite-measure continuity applied to $|\mu |$ proves $F_l(s)\to F_l(t)$ as $s\uparrow t$, and $F_r(s)\to F_r(t)$ as $s\downarrow t$. Moreover the opposite one-sided limits are $F_l(t+)=F_r(t)$ and $F_r(t-)=F_l(t)$. Consequently both representatives are continuous exactly where $\mu (\{t\})=0$. For each positive integer $m$, there are only finitely many measure atoms of magnitude at least $1/m$, since their total magnitudes are bounded by $|\mu |((a,b))$. Their union is countable, proving the requested discontinuity bound. This does not require the jump points to be isolated; they can be dense.

The hint's one-dimensional statement is consistent with this construction: zero ${\mathcal{H}}^0$ measure means an empty set, so $S_u\subseteq J_u$. This concerns approximate discontinuities of the BV space class; arbitrary changes of representative at points could create artificial pointwise discontinuities. The interval-mass representatives above remove that ambiguity and give the required one-sided continuity directly.