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?




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Does the estimate improve if we also assume $g$ to be invertible?




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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?



Estimate on total variation of composition of functions
If $f$ has a jump at 0, and $g:[0,1]\to\mathbb R$ crosses zero infinitely often, then var$(f\circ g)=\infty$.


