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Show integrable is uniformly continuous

  1. Apr 3, 2008 #1
    H = [a,b][tex]\times[/tex][c,d] . f:H[tex]\rightarrow[/tex]R is continuous, and
    g:[a,b][tex]\rightarrow[/tex]R is integrable.

    Prove that
    F(y) = [tex]\int[/tex]g(x)f(x,y)dx from a to b is uniformly continuous.

    I initially ripped g(x) and f(x,y) apart and tried to show each was continuous. This failed.
    In short, I am completely stuck. Please help me.
  2. jcsd
  3. Apr 3, 2008 #2


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    g(x) is only given as integrable, not necessarily continuous so that couldn't work.
  4. Apr 3, 2008 #3


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    Hint: since H is compact, f is, in fact, uniformly continuous. Use that to show for all epsilon>0, there exists a delta>0 such that |f(x,y0)-f(x,y)|<delta for all y such that |y-y0|<delta (for all x). Use that to show that the integral F(y) is continuous at y0. Once you know it's continuous, you don't have to worry about the uniform part, since y is in [c,d], which is also compact.
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