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)

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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	CATEGORIES ABOUT CLASSIFICATION OF DISCONTINUITIES

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Information About

Classification Of Discontinuities