Definition talk:Discontinuity of the First Kind

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Is it also necessary to state specifically that the limits from right and left are unequal? Or is it appropriate to leave it with the tacit understanding that if they were the same then the function would be continuous? --prime mover (talk) 09:59, 30 June 2016 (UTC)

Perhaps I am overtired so correct me on the following if I am wrong.
Equality of $f(x-)$ and $f(x+)$ (I assume this notation is understood by you) would not imply that the function is continuous at that point. See the function $f(x) = 1$ if $x = 0$, $f(x) = x$ else. This function is discontinuous at zero, though it certainly vanishes from either side. In general, this allows for removable discontinuities.
One may specify that $f(x-) \neq f(x+)$. This has the added benefit of differentiating the jump discontinuity from the removable discontinuity. One may wish to do this, as removable discontinuities are, more or less, trivial to deal with. Jump discontinuities that are not removable, may require a little more work.
On further thought, perhaps this definition should be changed to "discontinuity of the first kind", and jump discontinuity should have the added stipulation that $f(x-) \neq f(x+)$, so to give the graph that "jump" that one expects from the name. --Keith.U (talk) 13:31, 30 June 2016 (UTC)
An addendum: I've checked my source, and Rudin does refer to this general case (with removable discontinuities allowed) as a discontinuity of the first kind. However, other sources take a discontinuity of the first kind to not include removable discontinuities. I would put forth that the most unambiguous naming convention would to let a discontinuity of the first kind include removable discontinuities, and to let a jump discontinuity not include the removable case. --Keith.U (talk) 13:44, 30 June 2016 (UTC)
I know little about the details; such classification of minutiae tends to bore me. :-) But yes you are right that I forgot about the removable discontinuity.
If some sources say a jump discontinuity includes a removable discontinuity (i.e. that the latter is a j.d. such that f- = f+, excuse my laziness of terminology, you know what I mean) and some specifically exclude them, then this fact itself needs t be documented in an "also defined as" section (search through pw for examples of how this is used) with a warning to make sure the correct interpretation is used.
It may even be worth separating out the definition for "discontinuity of the first kind" into a separate page. If you are following the exposition as given in a particular source (this approach is highly recommended -- it promotes a consistent approach) then it is best to use the terminology found in that source unless it clashes with other terminology found on this site, in which case a compromise will need to be sought. As the Rudin source has already been used by a few contributors who have posted up a few random disconnected results from it, it may be a profitable approach to work through that source methodically and ensure that everything posted up from there depends logically upon something else from there. Picking results and definitions from the middle of a work can leave such matters unresolved and thereby causing ambiguity and inaccuracy in $\mathsf{Pr} \infty \mathsf{fWiki}$. --prime mover (talk) 14:42, 30 June 2016 (UTC)