Thread: Continuity with Integrals

1. Continuity with Integrals

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

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

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

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

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

4. 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 $\displaystyle f$ is differentiable and thus trivially continuous, no?

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