Let $f$ be a function of bounded variation on $[a, b]$ and $T_{a}^{b}(f)$ its total variation. We do not assume that $f$ is continuous. Show that $$\int_{a}^{b}|f'(t)|\, dt \leq T_{a}^{b}(f).$$
I know that if we assume that $f$ is continuous, then the above equation is true because we have the ability to use the Mean Value Theorem. What can I do if we don't assume $f$ is continuous?
Best Answer
Since $f$ is of bounded variation, we can write $f = g -h$ where $g$ and $h$ are monotone increasing and $f' = g' -h'$ a.e. and $$|f'| \leq |g'| + |h'| = g' + h'.$$ So
$$ \int_a^b |f'| \leqslant\int_a^b g' + \int_a^b h' \leqslant \ldots $$
Can you take it from here?