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This is a follow-up to a question asked by a calculus student here: Is the function $\frac{x^2-x-2}{x-2}$ continuous?

It got me thinking about a more interesting related question in classical analysis: When in general is a discontinuity for a real valued map removable? A rational function with a factorable polynomial in the numerator with a factor in the denominator is an almost trivial case - but when is it true in general? In complex analysis, for holomorphic functions, there's a very well-defined theory based on a theorem of Riemann. But what about real valued maps?

While I'm on the subject, I should note that "removable discontinuity" in the sense used in this problem is something of a misnomer - the point really should be called a removable singularity. The technical distinctions are nicely summed up here:

http://en.wikipedia.org/wiki/Classification_of_discontinuities

Addendum: I just found this older,related post at this board: Is there a function with a removable discontinuity at every point? If this is a valid algorithm, it may provide a starting point for a theoretical basis for the solution to my question!

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    When the left and right limits of the function at the singularity exist and agree?2011-11-26
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    You can't make a general statement for continuity without prooving the limits of the function in the discontinuity-point. The general statement is the existence of the limit. Or didn't I get your point?2011-11-26
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    @Mathemagician1234: Your comment was off-topic. Please either calmly ask "Why the downvote?", or don't say anything about it.2011-11-26
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    @Zev Got it-sorry.I'm just losing my patience with it.2011-11-26
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    @user7530, Daniel: I'm not even sure the question makes sense if the left and right hand limits around the singularity don't agree. The 2 sided limit exists or whether or not we can remove the singularity is irrelevant-we CAN'T make it continuous at a point where the one sided limits don't agree,yes?2011-11-26
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    I wasn't the first downvote, but I just added a second downvote. Given your self-reported background, this question is really silly and trivial. A version of it appears as an exercise in every textbook I've ever used to teach freshman calculus.2011-11-26
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    @Mathemagician1234: Your comment does not seem to me to make a lot of sense; the two-sided limit exists **if and only if** both one-sided limits exist and are equal. So "the 2-sidedd limit exists" is *exactly the same* as "the one sided limits exist and agree". How can the former be irrelevant, but the latter be the key, if they are equivalent?2011-11-26
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    @Arturo You misunderstood my response. I was agreeing with Daniel's assessment. It's important to make sure there's agreement on the obvious points-you never know which will be important in the solution of a problem.2011-11-27
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    @AdamSmith - The content of a question is what's important, not who asked it. If Terry Tao came in and asked a question you deemed below his station, would you downvote it? I cannot imagine any situation in which people should avoid asking a question when they don't know the answer - even if you think it "ought" to be within their ability, all *you* should worry about is giving a good answer for the benefit of the online math community as a whole.2011-11-29
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    I think it is clear that, while your comments on Mathemagician1234's posts in most cases do constitute valid criticisms, you have consciously focused on criticizing him in particular, and it has gotten to the point where the adversarial nature of your comments is doing more harm than good. At this time, I will *ask* you to let other users interact with Mathemagician1234 on occasions in which criticisms might be given to him, and only post yourself if some time has passed and no one has yet made the point you want to make (though, if that is the case, how important could it really be?)2011-11-29
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    But if necessary, the next step will be a "forced separation" (no interaction on the site, on pain of suspension).2011-11-29
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    I don't understand why this question got 6 downvotes. I would not upvote it but now I will to compensate for inappropriate downvotes. +12012-10-12

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