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Math Help - Question about continuous function

  1. #1
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    Question about continuous function

    A few questions about limits and continuity.

    1)Am I correct that for a continuous function, the definition of a limit allows x=a since abs(x-a) =0 -> abs(F(x)-F(a))=0< e?

    2) I have come across a function which I need to find the limit of at x=1. It requires lim(x^2+1) , x=1 when x^2+1 is not defined at x=1. Now I realise that for limits we are not interested what happens at x=1. It's clearly not continuous at x=1 so I seemingly cannot apply limit of F(x)=F(1). THe answer is obviously 2 but how do I justify that answer?

    Thanks for clearing up these simple questions.
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  2. #2
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    It doesn't need to be continuous. You SAY you are thinking about the neighborhood, but your wording suggests you are still thinking about the single value as if it were continuous.

    Move you mind away from the the point and out to the neighborhood. Consider the left and right limits. Do they exist? Are they the same?
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  3. #3
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    usually the definition of a limit of a function is stated like this:

    lim x-->a f(x) = L if for every ε > 0, there is some δ > 0 so that 0 < |x-a| < δ implies |f(x) - L| < ε.

    note that i wrote: 0 < |x-a| < δ, instead of |x-a| < δ. this allows for the fact that f(a) might not be defined.

    in other words, we consider "deleted neighborhoods" of a.

    if, in fact, f(a) DID exist, and was equal to L, the only way this can happen is if f is continuous.

    this is what makes continuous functions special: all limits exist, and we can just "substitute in a".

    for functions that are NOT continuous, it can still happen that lim x-->a f(x) will exist. an easy example is f(x) = x/x.

    lim x-->0 f(x) = 1, even though 0/0 is undefined. note that choosing δ = 0 would be a very poor choice,

    as we would be in the unfortunate situation of trying to prove |(0/0) - 1| < ε. on the other hand, choosing 0 < δ works very well:

    if 0 < |x| < δ, then |x/x - 1| = |(x-x)/x| = |0/x| = 0 < ε. note that it is important we NOT allow x to be 0, so that (x-x)/x is meaningful.

    you can break it down this way:

    a) does the limit exist? yes (f may be continuous), no (f is NOT continuous).
    b) is the limit equal to f(a)? yes (f is continuous), no (f is discontinuous at a).

    you can't do b) before a), you have to answer a) FIRST. if f(a) is undefined, then you won't ever get to b), but you can still answer a).
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