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Math Help - Integral and Continuity

  1. #1
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    Nov 2009
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    Unhappy Integral and Continuity

    Question Details:
    Hey guys,

    I'm in a pre-analysis course and I have this homework problem over a section that my professor has neglected to cover, so I am really lost. Any help would be appreciated.
    Here is the problem

    Let f(x,y) be defined and continuous for a \leq x \leq b and c \leq y \leq d., Let

    F(x)= \int_c^d f(x,y) \, dy
    Prove that F is continuous on [a,b].

    The book gave the hint "Use the uniform continuity of f." So I went through that since f is continuous and bounded, f is uniformly continuous. From there I'm not really sure where to go.

    Any help would be appreciated.

    Thanks
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  2. #2
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    Hint: |F(x)-F(x_0)|=|\int f(x,y)-f(x_0,y) dy|\leq \int |f(x,y)-f(x_0,y)| dy\leq \int \epsilon dy.
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