Let $f(t,v)$, $t\in[0,T]$, $v\in R$ be absolutely continuous in $t$ for every fixed $v$ and absolutely continuous in $v$ for every fixed $t$. Let $X(t)$, $t\in [0,T]$ be a continuous function of unbounded variation on every interval $[t_1,t_2]\subset[0,T]$. I propose that if for every $t$ $f'_v(t,v)\neq0$ a.e. then $f(t,X(t))$ in a function of unbounded variation.
Need help in proving this. I'll be thankful for any textbook recommendations on related topics.