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Math Help - Uniform Continuity

  1. #1
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    Uniform Continuity

    prove g(x)=x^2 + 2x - 5 is uniformly continuous on [0,3].

    proof: Let e>0 and choose delta=e/3. If |y-x|< delta and x,y e[0,3] then |g(y)-g(x)| = |(y^2 +2y -5)-(x^2 +2x -5)|=|y^2 -x^2 +2y-2x|=
    |y(y+2) - x(x+2)| < 3|y-x| < (3)delta = e.

    Did i do the proof right? I get confused when an interval is given such as [0,3]. What is the difference if the interval was [0,3)? How does that effect the proof?
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  2. #2
    Member Black's Avatar
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    I don't think that inequality is going to work (take x=2, y=3).

    Ok, so we have

    |y^2+2y-5-x^2-2x+5|=|y^2-x^2+2y-2x|=|(y+x+2)(y-x)|.

    Since 8 is the maximum number you can get out of y+x+2 on the interval [0,3], let \delta=\varepsilon/8.
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  3. #3
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    Quote Originally Posted by summerset353 View Post
    Did i do the proof right? I get confused when an interval is given such as [0,3]. What is the difference if the interval was [0,3)? How does that effect the proof?
    being continuous on a compact set, then it's uniformly continuous, and obviously uniformly continuous on [0,3) since this set is a subset of [0,3].
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