Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets. Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function. Let $s:Y\rightarrow X$ be a function (not necessarily continuous). Define $m:X\times\mathbb{R}\rightarrow\mathbb{R}$ as: $m(x,h)=\int_{S(x,h)}f(x+h,y)dy$

where $S(x,h)= [ y \in Y:x \leq s(y) < x+h ] $ with $h>0$ and small.

Finally, $\forall(x,y)\in X\times Y$ such that $s(y)=x,f(x,y)=0$.

Question: Calculate the limit as $h\rightarrow0$ of $m(x,h)$