Happy Thanksgiving to you all.
I have made an attempt to a homework problem, that I'll need someone look over for me. The question is a follows.
If $f$ is of bounded variation on $[a,b]$ then $f'(x)$ exists a.e. and$$\int_a^b |f'|~dx\leq T_a^b(f),$$ where $T(f)$ is the total variation.
This is my attempt.
Since $f$ is of bounded variation, we can write $f(x)=g(x)-h(x)$ where $g$ and $h$ are monotone increasing functions on $[a,b]$. Thus $f'(x)=g'(x)-h'(x)$ a.e. Furthermore, $|f(x)|\leq |g'(x)|+|h'(x)|=g'(x)+h'(x)$. So $$ \begin{align*} \int_a^b|f'(x)|~dx & \leq \int_a^b g'(x) + \int_a^b h'(x)\\
& \leq g(b)-g(a)+h(b)-h(a)\\
& = T_a^b(f).
\end{align*}
$$ I'm also required to find additional conditions on $f$ needed for equality to hold.
Is it enough to say that equality holds if $f$ is absolutely continuous, or must I justify it?