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Math Help - Question regarding continuity

  1. #1
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    Question regarding continuity

    I have a couple of questions regarding continuity. One is just a general question and the second is a textbook question.

    1) Can anyone give me a quick explanation in regards to the concept of continuity? We transitioned from limits (which I picked up really well) into continuity and I'm having a tough time distinguishing between the two.

    2) The directions state: Find all values of x = a where the function is continuous. For each value of x, give the limit of the function as x approaches a.

    The Problem is: f(x) = 5 + x/x(x-2)

    I'm ending up with 5 + x/-2, but I'm completely lost as to the next step in the problem.

    Any help would be greatly appreciated. Thanks!
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  2. #2
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    Re: Question regarding continuity

    A function is continuous at a point x=a if the function evaluated at a, i.e. f(a), equals both the limit of f(x) as x approaches a from the left and as x approaches a from the right

    In math symbols, the function is continuous at a if
    f(a)=\lim_{x\rightarrow a^-} f(x) =\lim_{x\rightarrow a^+} f(x)
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  3. #3
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    Re: Question regarding continuity

    continuity and limits are closely linked.

    as the previous poster noted, f is continuous at a if lim x-->a f(x) = f(a).

    this means two things:

    1) the limit lim x-->a f(x) has to exist (the right-hand and left-hand limits must both exist, AND be equal).

    2) f(a) has to exist.

    for example, the function f(x) = x/x, has a perfectly good limit at x = 0: lim x-->0 x/x = 1.

    however, the function is NOT continuous there, as f(0) is undefined (0/0 has no well-defined meaning).

    now what it means for lim x-->a f(x) = f(a) is this: as we get close to a, f doesn't do anything unexpected, such as have a hole, or suddenly switch values, or become infinitely large in size.

    put another way: if x is near a, then f(x) is near f(a) (the epsilons and deltas are for making precise this notion of "near").

    your function (which i hope is) f(x) = -5 + x/(x(x-2)), has two discontinuities: one at x = 0 (it has a "hole" there, because it is undefined, even though the limit lim x-->0 f(x) exists),

    and another at x = 2, where f becomes unbounded (this is another way of saying it becomes "infinitely big" in size).
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