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 <math>f(x)</math> of real variable <math>x</math> that is defined for <math>x</math> to the left and to the right of a given point <math>x_0</math>, that is, for <math>x<x_0</math> and <math>x>x_0</math>. Then three situations are possible:
1. The one-sided limit from the negative direction
- <math>L^{-}=\lim_{x\rarr x_0^{-}} f(x)</math>
and the one-sided limit from the positive direction
- <math>L^{+}=\lim_{x\rarr x_0^{+}} f(x)</math>
at <math>x_0</math> exist, are finite, and are equal. Then, x0 is called a removable discontinuity.
2. The limits <math>L^{-}</math> and <math>L^{+}</math> exist and are finite, but not equal. Then, x0 is called a jump discontinuity.
3. One or both of the limits <math>L^{-}</math> and <math>L^{+}</math> does not exist or is infinite. Then, x0 is called an essential discontinuity.
Examples
1. Consider the function
- <math>f(x)=\left\{\begin{matrix}x^2 & \mbox{ for } x< 1 \\ 2-x& \mbox{ for } x>1\end{matrix}\right.</math>
Then, the point <math>x_0=1</math> is a removable discontinuity. One can make this function continuous by setting <math>f(x_0)=1.</math>
2. Consider the function
- <math>f(x)=\left\{\begin{matrix}x^2 & \mbox{ for } x< 1 \\ 2-(x-1)^2& \mbox{ for } x>1\end{matrix}\right.</math>
Then, the point <math>x_0=1</math> is a jump discontinuity.
3. Consider the function
- <math>f(x)=\left\{\begin{matrix}\sin\frac{5}{x-1} & \mbox{ for } x< 1 \\ & \\ \frac{0.1}{x-1}& \mbox{ for } x>1\end{matrix}\right.</math>
Then, the point <math>x_0=1</math> 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.
