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Real Analysis: Continuity and Uniform Continuity

  1. Jul 20, 2011 #1
    Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous?


    My attempt: So I know that continuity is defined as
    "given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then abs(f(x)-f(y))<Epsilon.

    So i tried expanding the function, but still can not find the values for delta that make this continuous on [0,infinity). Any ideas?

    Also, it is NOT uniformly continuous correct?

    Thanks!
     
  2. jcsd
  3. Jul 20, 2011 #2

    lanedance

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    well its definitely continuous on [0,inf) - can you show that?

    now for the "uniform continuity part", not as the actual proof, but I would consider the derivative - it should give you an idea of how to approach the problem

    loosely speaking if a function is uniformly continuous it shouldn't have a derivative that misbehaves

    consider
    [tex] |\frac{f(y)-f(x)}{y-x}|[/tex]
    in the limit y goes to x, this becomes the derivative...
     
  4. Jul 20, 2011 #3
    Hey, so one of the problem I had was that I could not find a way to show it was continuous. I tried to show it using the delta-epsilon definition of continuity, but couldnt figure out a way to find delta.
     
  5. Jul 20, 2011 #4

    lanedance

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    ok so where did you get stuck?
     
  6. Jul 20, 2011 #5
    So i did absolute(f(x)-f(y))= abs( ((x^2)/((x^2)+1)) - ((y^2)/((y^2)+1)) ). I simplified this to get abs( ((x^2)-(y^2))/((x^2)+1)((y^2)+1)) ).

    Now I tried to set it up so that abs ( x-y ) is less than 1 (which would be one of the minimum components of delta) so that I can try to abs( x-y ) less than something with respect to epsilon. However, I am stuck on how I can use <1 to find the deltas that make abs( f(x)-f(y) ) < E for any given E.
     
  7. Jul 20, 2011 #6
    I believe you have your definitions backwards. A function is continuous at a point x if given epsilon there is a delta. If you pick some different point y, then given the same epsilon, you may need a different delta.

    If the same delta works for a given epsilon regardless of x, that is uniform continuity.

    What you wrote above is the definition of uniform continuity ... given epsilon, for any x there's a delta. The def of continuity is that given x, for any epsilon there is a delta that depends on x.
     
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