Classification of discontinuities

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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 x0, that is, for x < x0 and x > x0. Then three situations are possible:

1. The one-sided limit from the negative direction

L^{-}=\lim_{x\rarr x_0^{-}} f(x)

and the one-sided limit from the positive direction

L^{+}=\lim_{x\rarr x_0^{+}} f(x)

at x0 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.

[edit] Examples

The function in example 1, a removable discontinuity
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The function in example 1, a removable discontinuity

1. Consider the function

f(x)=\begin{cases}x^2 & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ 2-x& \mbox{ for } x>1\end{cases}

Then, the point x0 = 1 is a removable discontinuity. One can make this function continuous by setting f(x0) = 1.

The function in example 2, a jump discontinuity
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The function in example 2, a jump discontinuity

2. Consider the function

f(x)=\begin{cases}x^2 & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ 2-(x-1)^2& \mbox{ for } x>1\end{cases}

Then, the point x0 = 1 is a jump discontinuity.

The function in example 3, an essential discontinuity
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The function in example 3, an essential discontinuity

3. Consider the function

f(x)=\begin{cases}\sin\frac{5}{x-1} & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ \frac{0.1}{x-1}& \mbox{ for } x>1\end{cases}

Then, the point x0 = 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.

[edit] External links

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