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
