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. The oscillation of a function at a point quantifies these discontinuities as follows: A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity). For each of the following, consider a real valued function of a real variable , defined in a neighborhood of the point "x"0 at which is discontinuous. Consider the function   The point = 1 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 = = . In other words, since the two one-sided limits exist and are equal, the limit of as approaches exists and is equal to this same value. If the actual value of is "not" equal to , then is called a removable discontinuity. This discontinuity can be removed to make continuous at , or more precisely, the function  is continuous at = . The term "removable discontinuity" is sometimes an abuse of terminology for cases in which the limits in both directions exist and are equal, while the function is undefined at the point . This use is abusive because continuity and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain is properly named a removable singularity. Consider the function   Then, the point = 1 is a "jump discontinuity". In this case, a single limit does not exist because the one-sided limits, and , exist and are finite, but are not equal: since, ≠ , the limit does not exist. Then, is called a "jump discontinuity", "step discontinuity", or "discontinuity of the first kind". For this type of discontinuity, the function may have any value at . For an essential discontinuity, only one of the two one-sided limits need not exist or be infinite.