# Discontinuity point

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A point in the domain of definition of a function , where and are topological spaces, at which this function is not continuous. Sometimes points that, although not belonging to the domain of definition of the function, do have certain deleted neighbourhoods belonging to this domain are also considered to be points of discontinuity, if the function does not have finite limits (see below) at this point.

Among the points of discontinuity of a function, defined on deleted neighbourhoods of points on the real axis, one distinguishes points of the first and second kind. If a point is a point of discontinuity of a function that is defined in a certain neighbourhood of this point, except perhaps at the point itself, and if there exist finite limits from the left and from the right for (in a deleted neighbourhood of ), then this point is called a point of discontinuity of the first kind and the number is called the jump of at . If moreover this jump is zero, then one says that is a removable discontinuity point. If the discontinuity point is not of the first kind, then it is a discontinuity point of the second kind.