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The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
This definition allows a limit to be defined at limit points of the domain S, if a suitable subset T which has the same limit point is chosen. Notably, the previous two-sided definition works on int S ∪ iso S c , {\displaystyle \operatorname {int} S\cup \operatorname {iso} S^{c},} which is a subset of the limit points of S .
A removable discontinuity occurs when () = (+), also regardless of whether () is defined, and regardless of its value if it is defined (but which does not match that of the two limits). A type II discontinuity occurs when either f ( c − ) {\displaystyle f(c^{-})} or f ( c + ) {\displaystyle f(c^{+})} does not exist (possibly both).
A graph of a parabola with a removable singularity at x = 2 In complex analysis , a removable singularity of a holomorphic function is a point at which the function is undefined , but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than (hence a set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.
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This page was last edited on 10 January 2015, at 10:07 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply.