Talk:Classification of discontinuities

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===Better to classify like this:

==1) Removable

==2) Essential ... which have the following sub classifications....

=2a) (Finite) Jump =2b) Infinite (Jump) =2c) Oscillatory


==1) means... limit exists.... but not equal to function value

==2) means....limit does not exist

=2a) limit does not exist because the left and right limits (which exist and are finite) are not equal =2b) limit does not exist because left and/or right limit is +/- infinity =2c) limit does not exist because of oscillatory action near point --anon

I don't think that a finite jump discontinuity is an "essential discontinuity". Do you have references for that? Oleg Alexandrov (talk) 13:59, 15 May 2006 (UTC)
(Finite) Jump Discontinuities are essential because the criteria to determine if a discontinuity is essential or not is whether the limit exists at the point. Of a function has a finite jump, the limit does not exist there (left and right limits are unequal), thus there exists an essential discontinuity there.
here is the first link I clicked after searching for essential disc http://oregonstate.edu/instruct/mth251/cq/Stage4/Lesson/jumps.html
Unfortunately, this use of essential clashes with the use of essential singularity in complex analysis. The same goes for the article, by the way. Another possible point of confusion with the classification in the article is that "not removable" and "non-removable" are not the same. -- Jitse Niesen (talk) 02:55, 17 May 2006 (UTC)

[edit] Confusion

This article was confusing continuous functions (which are continuous on their domains) with continuous funciton on the real numbers. So technically all the example functions are continuous if they are not defined at x=1. I have edited the examples so that they are all total functions, removing this confusion. I would like to make the jump discontinuity half-continuous but then I would have to edit the image. —The preceding unsigned comment was added by 2006 80.57.33.218 (talk • contribs) .

You are right of course. I did not think of that when i added the examples. Oleg Alexandrov (talk)

[edit] Removal of isolated point case

I removed the following paragraph by Patrick

Note that a function is continuous if it is continuous in all the points of its domain. If a continuous function is not defined in some isolated point, we can similarly distinguish the three cases. While above "removable discontinuity" means that we can make the function continuous at the point by changing the function value, if the function is not defined yet the corresponding case means that we can make the function continuous at the point by defining the function value.

The issue of whether the function is actually defined at x0 is relevant only for removable discontinuities, and that case is already treated by the remark added by Patrick at that case. Repeating the same thing again in a separate paragraph does not make sense to me. Oleg Alexandrov (talk) 04:51, 28 May 2007 (UTC)

Never mind. I think Patrick has a point, although it took me a long time to understand what the point was from the text above. I tried to add something to that extent in the article, but stated differently. Oleg Alexandrov (talk) 05:16, 28 May 2007 (UTC)
After having slept on it, I believe if a function has a removable discontinuity at a point, it has to have a discontinuity there, so it has to be defined there. As such, it does not make sense to talk about things if the function is not defined at the given point, per the anon remark in the previous section. For that reason I removed again the case when the function is not defined at a point but is defined around it. Oleg Alexandrov (talk) 15:28, 28 May 2007 (UTC)
I agree that "make the function continuous at the point by defining the function value." was not correct, that should have been "defining the function value such that the function is continuous at this point". That is an interesting property (e.g. of (sin x)/x at x=0) that could be mentioned on this or another page.--Patrick 14:16, 29 May 2007 (UTC)

The last paragraph under the first section used to read like this:

The term removable discontinuity is sometimes (improperly) used for cases in which the limits in both directions exist and are equal, while the function is undefined at the point x0.[1] This use is improper because, according to the precise definition of continuity of a function, the function is then actually continuous.

I changed the last sentence of this paragraph to read as follows:

The term removable discontinuity is sometimes (improperly) used for cases in which the limits in both directions exist and are equal, while the function is undefined at the point x0.[2] This use is improper because the precise definition of continuity of a function applies only to points which are in the domain of the function.

As I understand it, the concept of continuity is defined only for points in the domain of the function, so for points outside the domain of the function it doesn't make sense to say that the function is either continuous or discontinuous. I changed the wording of the sentence to make this clearer, I think, and also because it is not necessarily true that a function is continuous if the limits in both directions exist and are equal and the function is undefined at the point. For example, consider the function f\colon\mathbb{R}\setminus\{0\}\to\mathbb{R} defined by

f(x)=\begin{cases}x,&\mbox{if }x\mbox{ is rational};\\0,&\mbox{if }x\mbox{ is irrational}.\end{cases}

Then \lim_{x\to0^+}f(x)=\lim_{x\to0^-}f(x)=0, but the function is continuous nowhere in its domain. —Bkell (talk) 05:45, 10 October 2007 (UTC)

A common way of making functions total is to turn the codomain into a pointed set by adding one new element standing for "undefined", often represented by the symbol ⊥. If the codomain C is a topological space, then so is the disjoint union C ∪ {⊥} of C and {⊥} viewed as terminal object of Top. Then the "reparable holes" as for sin(x)/x become removable discontinuities. Specialized to the case of functions defined on real numbers, it is possible to define an operation that takes a function f to the function g defined by:
g(x)=\begin{cases}y,&\mbox{if }\lim_{\xi\to x^-}f(\xi)=y=\lim_{\xi\to x^+}f(\xi);\\f(x),&\mbox{if no two-sided limit exists}.\end{cases}
This operation will remove removable discontinuities and fix reparable holes. One would expect such an operation to have been discussed in the literature – although I'm unaware of any instances.  --Lambiam 11:25, 10 October 2007 (UTC)

Calling the use of the term removable discontinuity improper is POV. The use of the word discontinuity to refer to points outside the domain of a function is well-established. James Stewart gives the following definition:

If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a.

David Radcliffe (talk) 20:52, 3 May 2008 (UTC)

In my opinion it is improper to use a term for instances that are not covered by its definition. The function defined by f(x) = (sin x)/x is not defined in any open interval containing 0. Therefore the definition does not apply, and it is improper to state that the function has a removable discontinuity for x = 0. Why is that POV?  --Lambiam 12:22, 4 May 2008 (UTC)