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Math Help - Uniformly Continuous Problem

  1. #1
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    Uniformly Continuous Problem

    Q: Let f:[0,infinity) be defined by f(x) = x^2/(x+1). Prove if it is uniformly continuous.

    My solution: I let \epsilon > 0, \delta > 0, x,y\epsilon[0,infinity) with |x-y|<\delta

    then I have |f(x)-f(y)| =  |\frac{x^2}{x+1}-\frac{y^2}{y+1}| = |\frac{x^2+x^2y-y^2x-y^2}{(x+1)(y+1)}|

    But I can't simpify it anymore to which would make the proof works...

    Thank you.
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  2. #2
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    Have you noticed that the derivative is bounded?
     f'(x) = \frac{{x^2  + 2x}}{{x^2  + 2x + 1}} \le 1\quad x \in [0,\infty ).
    Then we can get \left| {f(x) - f(y)} \right| \le \left| {x - y} \right|
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  3. #3
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    I understand that the derivative is bounded, but how does that imply |f(x)-F(y)| < |x-y| ? I have not learn to do anything with derivative in my class yet.
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  4. #4
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    Quote Originally Posted by tttcomrader View Post
    I understand that the derivative is bounded, but how does that imply |f(x)-F(y)| < |x-y| ? I have not learn to do anything with derivative in my class yet.
    Sorry to say, someone has gotten the cart before the horse!
    This is a classical problem.
    It is a Lipschitz Condition. The derivative is bounded.
    That implies uniform continuity.
    You need to confront your lecturer.
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