Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is $$ \int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ? $$ For example is it true that $$ \int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \le TV(f) \Vert g' \Vert_{L^\infty} $$ holds?
Does the estimate improve if we also assume $g$ to be invertible?
As a bonus question: if we additionally assume that $g$ is a homeomorphism, is it true that $$ \int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx = TV(f) $$ hold?