This question was answered in the [comments](https://mathoverflow.net/questions/83545/variation-of-function#comment215095_83545) by [fedja][1]. Consider 
$$
f(t,v)=\min_s \sqrt{(X(s)-v)^2+(t-s)^2}.
$$ 
This function is Lipschitz, as the distance from $(t,v)$ to the graph of $X$. Whether or not $X$ has bounded variation, $f(t,X(t))$ is identically zero.


  [1]: https://mathoverflow.net/users/1131/fedja