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Math Help - Discontinuous function

  1. #1
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    Discontinuous function

    Hi,

    I have a piece-wise function similar to this:

    f(x) = (-ln x)/3 (x>0)
    -2/x (x<0)

    The limit of this is infinity, it exists as the one sided limits exist for f(x) and both equal infinity (although can it be argued that lim x->0 -2/x DNE?).

    So how does one classify this discontinuity, is it a removable or essential discontinuity? Why?

    Any help is appreciated.

    Cheers
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  2. #2
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    Re: Discontinuous function

    Quote Originally Posted by iMagoo View Post
    Hi,

    I have a piece-wise function similar to this:

    Code:
    f(x) = (-ln x)/3 (x>0)
          -2/x (x<0)
    The limit of this is infinity, it exists as the one sided limits exist for f(x) and both equal infinity (although can it be argued that lim x->0 -2/x DNE?).
    So how does one classify this discontinuity, is it a removable or essential discontinuity? Why?
    What definition of a removable and/or essential discontinuity do your notes/textbook use?

    In most cases it is a removable only if it is possible to redefine the function so as to have a finite limit.
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  3. #3
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    Re: Discontinuous function

    If lim(x->a) f(x) DNE, essential discontinuity at a.
    If lim(x->a) f(x) = L =/= f(a) then it's removable.

    Turns out, underneath the limit laws, my book also defines: lim (x->a) f(x)/g(x) = L/M if M=/=0, DNE if M=0, L=/=0.

    Using this definition, lim (x->0) -2/x DNE, therefore two sided limits are not equal, therefore the discontinuity is essential. Is this correct?
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