Question

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

Answer 1

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_(xrarra)f(x)` exists, then `f` has a removable discontinuity at `a`
("Infinite limits" are "limits" that do not exists.)

We remove the discontinuity by defining:

`g(x) = {(f(x),"if",x != a" and "x in "domain"(f)),(lim_(xrarra)f(x),"if",x=a) :}`