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Math Help - Continuity with Integrals

  1. #1
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    Continuity with Integrals

    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...
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  2. #2
    Super Member girdav's Avatar
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    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].
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    Im really sorry but I have no idea how you know that or how that helps me...
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    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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