Continuity with Integrals

**zebra2147** · March 9th 2011, 09:34 AM

If anyone can give me a push-start on the following proof I would appreciate it:

Suppose $g:\mathbb{R}\times [a,b]\rightarrow \mathbb{R}$ is continuous. Show $f:\mathbb{R} \rightarrow \mathbb{R}$ determined by
$f(x)=\int _{a}^{b}g(x,y)dy$ is continuous.

I know that $f'(x)=\int _{a}^{b}g_x(x,y)dy$ but I'm not sure if that helps us here...

**girdav** · March 9th 2011, 09:57 AM

Fix $x_0\in\mathbb R$ . $g$ is uniformly continuous on the compact set $\left[x_0-1,x_0+1\right]\times \left[a,b\right]$ .

**zebra2147** · March 9th 2011, 04:30 PM

Im really sorry but I have no idea how you know that or how that helps me...

**Drexel28** · March 9th 2011, 07:21 PM

Originally Posted by zebra2147

Im really sorry but I have no idea how you know that or how that helps me...

I must be misunderstanding. If you know the result about Leibniz's differentiation under the integral sign you must know then that $f$ is differentiable and thus trivially continuous, no?

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