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Let $F(x,y)=\int_a^x \int_c^y f(s,t)dsdt$ for $(x,y)\in D:=\{(x,y):a\leq x \leq b, c\leq y\leq d \}$, where $f$ is integrable on $D$.

Is it true that $F_{xy}, F_{yx}$ exist a.e in $D$ and $F_{xy}(x,y)=F_{yx}(x,y)=f(x,y)$ a.e. in $D$ ?


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Hint: Use Fubini's theorem to switch the order of integration. – Jeff Feb 2 '12 at 0:24

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