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Math Help - Real Analysis: Epsilon- Delta HELP NEEDED

  1. #1
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    Real Analysis: Epsilon- Delta HELP NEEDED

    I have a final on Friday and i'm still not sure about a lot of things for this class... Here's this question:

    let f: D-->R and define I f I: D--> R by I f I (x) = I f(x) I

    a) suppose that f: D-->R is continuous at cED. Prove using an epsilon-delta argument that I fI is continuous at c.
    b) Prove that I f I is continuous at c it does not necessarily follow that f is continuous at c.

    I am completely lost on this problem, my professor is horrible

    the I f I like represents absolute value? that's what all the I's stand for in the problem. Anyone know how to start this problem?
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  2. #2
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    Quote Originally Posted by Vitava61 View Post
    I have a final on Friday and i'm still not sure about a lot of things for this class... Here's this question:

    let f: D-->R and define I f I: D--> R by I f I (x) = I f(x) I

    a) suppose that f: D-->R is continuous at cED. Prove using an epsilon-delta argument that I fI is continuous at c.
    b) Prove that I f I is continuous at c it does not necessarily follow that f is continuous at c.

    I am completely lost on this problem, my professor is horrible

    the I f I like represents absolute value? that's what all the I's stand for in the problem. Anyone know how to start this problem?
    for part a)

    let \epsilon>0 be given.
    Since f is continous at c choose a delta

    |x-c|<\delta, |f(x)-f(c)|< \epsilon

    We wish to show that ||f(x)|-|f(c)||<\epsilon

    by the triangle inequality||a|-|b||<|a-b|

    ||f(x)|-|f(c)||<|f(x)-f(c)|<\epsilon

    QED.

    I will try to think of a counter example for b.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Vitava61 View Post
    b) Prove that I f I is continuous at c it does not necessarily follow that f is continuous at c.
    The |'s mean absolute values

    example: f(x) = \left \{ \begin{array}{rlr} 1 & \mbox{ if } x \ge 0 & \\ & & c = 0 \\ -1 & \mbox{ if } x < 0 & \end{array} \right.
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    Can you explain to me how that counter example proves that is I f I is continuous at c, it does not necessarily follow that f is continuous at c?

    Thanks for all your help, I've been able to either do, or make sense out of everything else.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Vitava61 View Post
    Can you explain to me how that counter example proves that is I f I is continuous at c, it does not necessarily follow that f is continuous at c?

    Thanks for all your help, I've been able to either do, or make sense out of everything else.
    the function i gave is not continuous at x = 0. As i hope is obvious.

    however, if you take the absolute value of the function, we have |f(x)| = 1. this is a polynomial and is continuous everywhere

    thus |f(x)| is continuous, but f(x) isn't
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