Total variation $t\mapsto \|f\|_{t,\operatorname{var}}$ of a cadlag function $f$ with finite variation is cadlag

Question

If $(X_t)_{t\in [0,T]}$ is cadlag and has finite variation, is it true that the total variation $t\mapsto \operatorname{Var}(X)_t$ is cadlag? ($\operatorname{Var}$ denotes the total variation)

This question is related. Erik has seen the proof concerning right-continuity. I want to see it since I couldn't figure it out myself. Can someone help? We're basically using this definition.

Answer 1

$t\mapsto\operatorname{Var}(X)_t$ is nondecreasing so it has left and right limits everywhere.

You can easily show with its definition that $$ \operatorname{Var}(X)_{t+}-\operatorname{Var}(X)_t=\vert X_{t+}-X_t\vert, $$ from which you trivially deduce that if $X$ is càd then so is $\operatorname{Var}(X)$.