= Càdlàg regularity of finite total variation
{title2=$\Delta V_t=|\Delta X_t|\quad(t>0)$}
If a <càdlàg function> has <finite variation> on compact intervals, its <total-variation process> is finite, nondecreasing, and <càdlàg>. Left limits follow from monotonicity. For right continuity, approximate the variation on $[t,s]$ by one finite partition. Changing its first endpoint from $t$ to $u\downarrow t$ changes the sum by at most $|X_u-X_t|$, which tends to zero. Additivity of variation then bounds $V(u)-V(t)$ by an arbitrarily small error. Its jump is the absolute jump of $X$. Without locally finite variation, right continuity can fail even as an extended-valued assertion: $X_t=t\sin(1/t)$ with $X_0=0$ has infinite variation on every interval starting at zero.
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