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 describes the classification of discontinuities in the simplest case of a function of a single real variable.

Consider a function $f(x)$ of real variable $x$ that is defined for $x$ to the left and to the right of a given point $x\_0$, that is, for

1. The one-sided limit from the negative direction

$L^\{-\}=\backslash lim\_\{x\backslash rarr\; x\_0^\{-\}\}\; f(x)$

and the one-sided limit from the positive direction

$L^\{+\}=\backslash lim\_\{x\backslash rarr\; x\_0^\{+\}\}\; f(x)$

at $x\_0$ exist, are finite, and are equal. Then, x₀ is called a removable discontinuity.

2. The limits $L^\{-\}$ and $L^\{+\}$ exist and are finite, but not equal. Then, x₀ is called a jump discontinuity.

3. One or both of the limits $L^\{-\}$ and $L^\{+\}$ does not exist or is infinite. Then, x₀ is called an essential discontinuity.

Examples

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1. Consider the function

Then, the point $x\_0=1$ is a removable discontinuity. One can make this function continuous by setting $f(x\_0)=1.$

2. Consider the function

Then, the point $x\_0=1$ is a jump discontinuity.

3. Consider the function

Then, the point $x\_0=1$ is an essential discontinuity. For it to be an essential discontinuity it would have sufficed that only one of the two one-sided limits did not exist or were infinite.

External links

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Mathematical analysis

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