# Definition talk:Discontinuity of the First Kind

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)
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)