Question

How do you remove a removable discontinuity?

Answer 1

Please see the explanation section.

Explanation:

Function `f` has a removable discontinuity at `x=a` if `lim_(xrarra)f(x) = L` (for some real number `L`)

But `f(a) !=L`

We "remove" the discontinuity at `a`, by defining a new function as follows:

`g(x) = { (f(x),if,x != a),(L,if,x=a) :}`

For all `x` other than `a`, we see that `g(x) = f(x)`. and `lim_(xrarra)g(x) = L = g(a)`

So `g` is continuous at `a`.

(In more ordinary language, g is the same as f everywhere except at x = a, and g does not have a discontinuity at a.)