# How can you remove a discontinuity?

Mar 11, 2018

#### Explanation:

A discontinuity at $x = c$ is said to be removable if

${\lim}_{x \rightarrow c} f \left(x\right)$ exists. Let's call it $L$.

But $L \ne f \left(c\right)$ (Either because $f \left(c\right)$ is some number other than $L$ or because $f \left(c\right)$ has not been defined.

We "remove" the discontinuity by defining a new function, say $g \left(x\right)$

by $g \left(x\right) = \left\{\begin{matrix}f \left(x\right) & \text{if" & x != c \\ L & "if} & x = c\end{matrix}\right.$.

We now have $g \left(x\right) = f \left(x\right)$ for all $x \ne c$ and $g$ is continuous at $c$,

Example

$f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$ is discontinuous at $x = 1$. ($f \left(1\right)$ does not exist)

But ${\lim}_{x \rightarrow 1} f \left(x\right) = {\lim}_{x \rightarrow 1} \frac{{x}^{2} - 1}{x - 1}$

$= {\lim}_{x \rightarrow 1} \left(x + 1\right) = 2$

So we remove the discontinuity by defining:

$g \left(x\right) = \left\{\begin{matrix}\frac{{x}^{2} - 1}{x - 1} & \text{if" & x != 1 \\ 2 & "if} & x = 1\end{matrix}\right.$.