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Math Help - Prove Function is Discontinuous Using Sequences

  1. #1
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    Prove Function is Discontinuous Using Sequences

    Hi I have to proove this function is discontinuous at x=0 using the formual definition of continuity:


    { 1 / squareroot(-x) x<0
    f(x) = { 0 x = 0
    { 1 / squareroot(x) x > 0

    Thanks a lot
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  2. #2
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    Use this sequence: a_n  = \frac{{\left( { - 1} \right)^n }}{{n^2 }}.
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    How did you find that sequence?
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    Quote Originally Posted by tbyou87 View Post
    How did you find that sequence?
    Thirty-five years of teaching the stuff.
    You need not understand how I found it.
    You must understand why it works as a counter-example.

    Do you see that the sequence converges to zero?
    But \lim _{n \to \infty } f\left( {a_n } \right) = \infty.
    What does that contradict about continuity?
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  5. #5
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    Yeah that makes sense. I don't know what I was thinking. I think in my head I wanted to do a proof as you would when you are proving that a function is continuous as opposed to discontinuous. But just to clarify my last question, in general it seems that any sequence of type (1/n^p) where p>0 or a^n where |a| < 1 are likely candidates for a contradiction.

    Here's my complete answer by the way:
    f(x0) = f(0) = 0
    (xn) = 1/n^2 subset of dom(f) = R
    (xn) -> 0 = x0 (can be easily proven using defenition of a limit)
    lim f(xn) = lim 1/squareroot(1/n^2) = +infinity does not equal f(x0) = f(0) = 0.

    You are basically looking for a sequence that goes to 0 but f(xn) goes to something other than 0 to prove discontinuity.

    Thanks that was very helpful. I've got another question this time on proving that a function is continuous. Your help was much appreciated. Thanks
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