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Math Help - How I find a Function f that is continuous but not uniformly continuous

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    How I find a Function f that is continuous but not uniformly continuous

    Let D be a nonemty set which is not closed. Find a function f: D->R such that f is continuous on D but not uniformly continuous on D.


    For this would X^2 be a function and how would I actually prove this?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by alice8675309 View Post
    Let D be a nonemty set which is not closed. Find a function f: D->R such that f is continuous on D but not uniformly continuous on D.

    For example f : ( 0 , 1 ] -> IR , f ( x ) = 1 / x
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    Quote Originally Posted by FernandoRevilla View Post
    For example f : ( 0 , 1 ] -> IR , f ( x ) = 1 / x
    Oh I see. But how would I prove that it satisfies the conditions :/
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by alice8675309 View Post
    Oh I see. But how would I prove that it satisfies the conditions :/

    It is continuous (why?) . You can prove that is is not uniformly continuous choosing epsilon = 10 . If there exists 0 < delta < 1 satisfyng the definition, take

    x = delta , y = delta / 11

    Then,

    | x - y | = ( 10 / 11 ) delta < delta

    and

    | f( x ) - f ( y ) | = ... = ( 10 / delta ) > 10

    which contradicts the definition.
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