Classification of discontinuities

In calculus, a branch of mathematics, a function is referred to there as everywhere discontinuous, where it is not continuous. A point where a function is discontinuous, we therefore also called a discontinuity or irregularity.

In the article, continuity is explained when a function is continuous and when it is discontinuous. In this article, different varieties (classes) are represented by discontinuities. Only real-valued functions on a real interval are considered.

Definition

As mentioned, is, a defined on the real interval function is discontinuous at the point if it is not constantly there. This is also called a discontinuous on a set function when the function is discontinuous at every point.

Classifications of discontinuities

We distinguish between different " varieties " of points of discontinuity. For this, the one-sided limits are considered:

For a real interval and you look at the location of the left-sided limit

And right hand limit

Now is continuous in, if both limits exist and are equal to the function value at the point. Otherwise is discontinuous at the location. The following cases are possible:

The cases 1 and 2 are also referred to as a first type of discontinuity; The cases 3 and 4. Accordingly as discontinuities of the second kind, or sometimes as an essential discontinuities

Examples

Example 1: The function

Has a removable discontinuity at the location.

Example 2: Function

Has a jump discontinuity at the point with a jump of 1

Example 3: Function

Has a discontinuity at the point of the second kind, the left-sided limit does not exist ( neither actually nor improperly), the right-hand limit.

Example 4: The Thoma ash function is discontinuous on the rationals and continuous on the irrationals. The Dirichlet function is discontinuous over its entire domain of definition.

Discontinuities of monotone functions

If the function on the real interval monotone, as exist for all the one-sided limits and. Therefore, such monotonic functions have no discontinuities of the second kind, the amount of points of discontinuity of the first kind of such monotonic functions is at most countable, but may well be dense in the domain.