Classification of discontinuities

Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous, or a function might not be continuous everywhere. If a function is not continuous at a point, one says that it has a discontinuity there. This article will describe the classification of discontinuities in the simplest case of a function of a single real variable.

Consider a function f(x) of real variable x which is defined to the left and to the right of a given point x₀. Then three situations are possible.

1. The limit from the left and the limit from the right at x₀ exist, are finite and equal. Then, x₀ is called a removable discontinuity.
2. The limit from the left and the limit from the right at x₀ exist and are finite, but not equal. Then, x₀ is called a jump discontinuity.
3. One of the limists from the left or from the right at x₀ does not exist or is infinite. Then, x₀ is called a essential discontinuity.