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Uniform continuity of composite function

  1. Apr 24, 2010 #1
    I'll be very thankful is someone will tell me where I'm wrong.

    We know:
    1) f is uniform continuous.
    2) g is uniform continuous.

    We want to prove:
    fg(x) is uniform continuous.

    from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e
    from 2 we know -> for every |x-y|<d exists |g(x)-g(y)|<d_0
    let a=g(x) and b=g(y) then
    for every x,y |x-y|<d exists |fg(x)-fg(y)|<e.

    Sorry for my poor formulation, English is not my mother tongue.
  2. jcsd
  3. Apr 24, 2010 #2
    What's wrong with it?
  4. Apr 24, 2010 #3
    I'll be more then happy if this is right...
    I'm just not sure.
  5. Apr 24, 2010 #4
    Yes, it's correct.
  6. Apr 24, 2010 #5

    You need to be careful to specify where (in their respective domains) f and g
    are uniformly continuous. Otherwise your statement is not true. I think Zhentil
    assumed f,g were everywhere unif. continuous.
  7. Apr 25, 2010 #6
    Oh, thanks for the remark.
    I indeed intended for every x,y in R.
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