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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 $f:D\rightarrow \mathbb{R}$ is integrable on $D$. Assume that $f$ is continuous at $(x_0,y_0)$.

Is it true that then $F_{xy}(x_0,y_0), F_{yx}(x_0,y_0)$ exist and $F_{xy}(x_0,y_0)=F_{yx}(x_0,y_0)=f(x_0,y_0)$ ?

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