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Math Help - any example of product of two uniformly continuous functions

  1. #1
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    any example of product of two uniformly continuous functions

    f,g are uniformly continuous on [a,b]

    any example f*g which is not uniformly continuous.
    and :

    example, f,g are bounded but fg is uniformly continuous.
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  2. #2
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    Quote Originally Posted by szpengchao View Post
    f,g are uniformly continuous on [a,b]

    any example f*g which is not uniformly continuous.
    and :

    example, f,g are bounded but fg is uniformly continuous.
    Is this what you meant to write?
    Anyway, if you're looking for functions f,g on [a,b] that are uniformly continuous and such that their product is not, you won't find any. Indeed, on a segment, uniformly continuous is equivalent to continuous (this is called Heine Theorem (at least in France)), and the product of two continuous functions is continuous.
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  3. #3
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    wrong

    Quote Originally Posted by Laurent View Post
    Is this what you meant to write?
    If you look for functions f,g on [a,b] that are uniformly continuous and such that their product is not, you won't find any. Indeed, on a segment, uniformly continuous is equivalent to continuous, and the product of two continuous functions is continuous. This is called Heine Theorem (at least in France).

    sorry.made a mistake. on subset X in R
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  4. #4
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    Quote Originally Posted by szpengchao View Post
    sorry.made a mistake. on subset X in R
    Check that x\mapsto x is uniformly continuous on \mathbb{R}, while x\mapsto x^2 is not.
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