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Classification Of Discontinuities




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 x and x>x_0. Then three situations are possible:

1. The '' One-sided Limit from the negative direction''
:L^{-}=\lim_{x arr x_0^{-}} f(x)
and the ''one-sided limit from the positive direction''
:L^{+}=\lim_{x arr x_0^{+}} f(x)
at x_0 exist, are finite, and are equal. Then, ''x''0 is called a removable discontinuity.

2. The limits L^{-} and L^{+} exist and are finite, but not equal. Then, ''x''0 is called a jump discontinuity.

3. One or both of the limits L^{-} and L^{+} does not exist or is infinite. Then, ''x''0 is called an essential or '''non-removable discontinuity'''.


EXAMPLES


1. Consider the function
:f(x)=\left\{\begin{matrix}x^2 & \mbox{ for } x< 1 \ 2-x& \mbox{ for } x>1\end{matrix} ight.
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
:f(x)=\left\{\begin{matrix}x^2 & \mbox{ for } x< 1 \ 2-(x-1)^2& \mbox{ for } x>1\end{matrix} ight.
Then, the point x_0=1 is a jump discontinuity.

3. Consider the function
:f(x)=\left\{\begin{matrix}\sin rac{5}{x-1} & \mbox{ for } x< 1 \ & \ rac{0.1}{x-1}& \mbox{ for } x>1\end{matrix} ight.
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.


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