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Mathematics-Online lexicon:

Removable Singularities

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A function $ f(x)$ has a removable singularity at the point $ x=x_{0}$ if the domain is a subset of $ D_{max}=\mathbb{R}\setminus \{x_{0}\}$ and if the limit

$\displaystyle \lim_{x \to x_{0}}f(x) $

exists.

One has to verify that the limit from the right coincides with the limit from the left.

$\displaystyle \lim_{x \to x_{0} \textnormal{, } x>x_{0}}f(x)\quad=\lim_{x \to x_{0} \textnormal{, } x<x_{0}}f(x)=c\qquad c \in \mathbb{R} $

Then the function has a removable singularity at the point $ (x_{0}/c)$. Usually one mark this removable singularity with an empty box in the sketch.

(Authors: Jahn/Knödler)
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  automatically generated 7/ 8/2004