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Let a function $f:[a,b)\times [c,d] \rightarrow \mathbb R$ be continuous with continuous $f_y$. Suppose that improper integral

$F(y)=\int_a^b f(x,y)dx$

is convergent for $y\in [c,d]$ and integral

$G(y)=\int_a^b f_y(x,y)dx$

is convergent locally uniformly, i.e.

integral $G$ is convergent for each $y\in [c,d]$ and moreover

for each $y_0 \in [c,d]$ and $\varepsilon >0$ there exist $\delta>0$ and $\eta >0$ such that $|\int_r^b f_y (x,y)dx|<\epsilon $ for each $r\in (b-\delta,b)$ and each $y\in (y_0-\eta, y_0+\eta)\cap [c,d]$.

Is it then true that

$$ \frac{d}{dy} \int_a^b f(x,y)dx=\int_a^b f_y(x,y)dx \ \ \ \ ? $$

Thanks

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