Classification of discontinuities
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a
discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the
classification of discontinuities in the simplest case of functions of a single real variable taking real values. Consider a real valued function
ƒ of a real variable
x, defined in a neighborhood of the point
x0 at which
ƒ is discontinuous. Three situations can be distinguished: ▪ The one-sided limit from the negative direction and the one-sided limit from the positive direction at exist, are finite, and are equal to. Then, if
ƒ is not equal to,
x0 is called a
removable discontinuity. This discontinuity can be 'removed to make
ƒ continuous at
x0', or more precisely, the function is continuous at
x=
x0. ▪ The limits and exist and are finite, but not equal.