A partial differential problem that I'm tripping over myself to do.

"Let $\Sigma$ be an open set in $\mathbb C^n$ and $S$ a compact set in $\mathbb R^m$. Let $f(x,y)$ be defined for $x\in\Sigma$, $y\in S$, and let $f$ be continuous as a function of $(x,y)\in\Sigma\times S$, and belong to $C^a(\Sigma)$ as a function of $x$ for any fixed $y\in S$. Show that$$F(x)=\int_Sf(x,y)\,dy_1\cdots\,dy_m$$belongs to $C^a(\Sigma)$."

Question

A partial differential problem that I'm tripping over myself to do.

"Let $\Sigma$ be an open set in $\mathbb C^n$ and $S$ a compact set in $\mathbb R^m$. Let $f(x,y)$ be defined for $x\in\Sigma$, $y\in S$, and let $f$ be continuous as a function of $(x,y)\in\Sigma\times S$, and belong to $C^a(\Sigma)$ as a function of $x$ for any fixed $y\in S$. Show that$$F(x)=\int_Sf(x,y)\,dy_1\cdots\,dy_m$$belongs to $C^a(\Sigma)$."

anonymous · Answer

@Limitless @FoolForMath @KingGeorge @TuringTest

Hehe, I'm dragging all of you in here..

anonymous · Answer

@eliassaab

anonymous · Answer

You too. >:D

zzr0ck3r · Answer

too many quantifiers for 2am:)

anonymous · Answer

Ahaha, I don't blame you.

anonymous · Answer

What do you mean by
$$
C^a(\Sigma )
$$

anonymous · Answer

I assume it's a scalar operating on the open set? Or maybe I have no idea what I'm talking about.

GRE reviews are all over the place.

anonymous · Answer

I think $ C^a(\Sigma )$ is the set of absolutely continuous function on $\Sigma$

anonymous · Answer

I can't find this notation in my PDE textbook. But that makes a lot of sense.

Sorry, I'm a bit sleepy.