How can you remove a discontinuity?

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Answer 1 · 2018-03-11T20:26:30

Please see below.

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

$lim_(xrarrc)f(x)$ exists. Let's call it $L$ .

But $L != f(c)$ (Either because $f(c)$ is some number other than $L$ or because $f(c)$ has not been defined.

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

by $g(x) = {(f(x),"if",x != c),(L,"if",x = c):}$ .

We now have $g(x) = f(x)$ for all $x != c$ and $g$ is continuous at $c$ ,

Example

$f(x) = (x^2-1)/(x-1)$ is discontinuous at $x=1$ . ( $f(1)$ does not exist)

But $lim_(xrarr1)f(x) = lim_(xrarr1)(x^2-1)/(x-1)$

$= lim_(xrarr1) (x+1) = 2$

So we remove the discontinuity by defining:

$g(x) = {((x^2-1)/(x-1),"if",x != 1),(2,"if",x = 1):}$ .