Discontinuous

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:

For each of the following, consider a real valued function f of a real variable x, defined in a neighborhood of the point x₀ at which f is discontinuous.

Consider the function

The point x₀ = 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 x₀ both exist, are finite, and are equal to L = L⁻ = L⁺. In other words, since the two one-sided limits exist and are equal, the limit L of f(x) as x approaches x₀ exists and is equal to this same value. If the actual value of f(x₀) is not equal to L, then x₀ is called a removable discontinuity. This discontinuity can be 'removed to make f continuous at x₀', or more precisely, the function

is continuous at x = x₀.

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