"Derivative" of piecewise Lipschitz continuous function

Question

Suppose that we are given a family of piecewise Lipschitz continuous functions $f^{\varepsilon}$ with one single jump at $x^{\varepsilon}$. We suppose that $x^{\varepsilon}$ is differentiable with respect to $\varepsilon$ at $\varepsilon=0$ and we denote its derivative by $\xi$. My question is why the following limit holds $$ \lim_{\varepsilon \to 0+}\frac{1}{\varepsilon}\int_{-\sigma}^{\eta} |f(x^0+\varepsilon \xi+y) - f(x^0+y) - \varepsilon \xi f_x(x+y)- (f(x-)-f(x+))\cdot \chi_{[-\varepsilon \xi,0]}(y)| dy =0. $$. Where $\sigma, \eta>0$ and $f(x \pm)$ denoting the right and left limit respectively. It seems to be like a Taylor-like expansion plus some additional jump condition. I'm interested how to proof that this limit holds, but I have no idea how to deal with the jump.