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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 $m(x,h)$

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  • $\begingroup$ Is this an exercise from a course? $\endgroup$
    – Yemon Choi
    Jan 29, 2013 at 22:12
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    $\begingroup$ Do you want to tell us why you're asking this question? (see the FAQ) $\endgroup$ Jan 29, 2013 at 22:13
  • $\begingroup$ Sandy, this website is for questions of research interest. Is there a research angle to your question? If not, it might fit better at math.stackexchange.com (but if you do post it there, be sure to explain how you came upon the question, why it interests you, what you know about it, what progress you've made on it, where you get stuck, and so on). $\endgroup$ Jan 29, 2013 at 22:24
  • $\begingroup$ Yemon: it is not an exercise. I am a phd student in quantitative sociology and it is where a model that I wrote done led me... Anthony: I am bit rusty with convergence of sets and don't really know where to look for an answer. Gerry: this is of research interest to me. but this might be indeed a bit easy for mathematicians hihi. $\endgroup$
    – Sandy
    Jan 29, 2013 at 23:00
  • $\begingroup$ Mathematical questions from other research areas are usually well-accepted here, including questions with mathematical content of elementary level. However, for the sake of a profitable exchange, some explanation and motivation about the problem is welcome. $\endgroup$ Jan 30, 2013 at 3:42

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