# What is the difference between a removable and non-removable discontinuity?

Nov 9, 2015

See the explanation below.

#### Explanation:

Geometrically, a removable discontinuity is a hole in the graph of $f$.

A non-removable discontinuity is any other kind of discontinuity. (Often jump or infinite discontinuities.)

Definition

If $f$ has a discontinuity at $a$, but ${\lim}_{x \rightarrow a} f \left(x\right)$ exists, then $f$ has a removable discontinuity at $a$
("Infinite limits" are "limits" that do not exists.)

We remove the discontinuity by defining:

$g \left(x\right) = \left\{\begin{matrix}f \left(x\right) & \text{if" & x != a" and "x in "domain"(f) \\ lim_(xrarra)f(x) & "if} & x = a\end{matrix}\right.$