Total variation of a cadlag function

Question

Let $f: [0, +\infty) \rightarrow \mathbb{R}$ be a cadlag function with $\|f \|_{t, \text{var}} < +\infty$, where $\|f \|_{t, \text{var}} < +\infty$ denotes the total variation of $f$ over the interval $[0,t]$.

There is an exercise asking us to prove that the function $g: [0, \infty) \rightarrow \mathbb{R}; \, t \mapsto \|f \|_{t, \text{var}}$ is also cadlag. (However, I have only seen the proof concerning right-continuity.)

Also, how can we compute the jump of the total variation function? (i.e. express $g(t) - g(t-) $ in terms of $f$)

Answer 1

Hint: Fix $t_0\in [0,\infty)$. The same argument which is used to establish the right continuity of $g=\|f\|_{t,var}$ at $t_0$ can be used to establish the left continuity of $\|f_0\|_{t,var}$ at $t_0$, where $f_0$ is defined to be $f$ except at $t_0$ where it is defined as $\lim_{t\to t_0^-}f(t)$. Then you can show that $g(t)-g(t^-)=|f(t)-f(t^-)|$.