I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.
For the first part: wikipedia has a proof. So, I am just posting the link here.
For the second part: are you considering functions of bounded variation on some interval? If so, then such a function can be written as the difference of two non decreasing functions, and hence is in $L^\infty$. So, this is answered by the first case.