A removable discontinuity is one at which the limit of the function exists but does not equal to the value of the function at that point.

A non removable discontinuity is any discontinuity which is not removable.

(a) A function with a non removable discontinuity at .

Let the function is .

 is undefined at .

(b) A function with a removable discontinuity at .

Let the function is .

The denominator of the function should not be zero.

.

The function has discontinuity at .

Rewrite the function as .

Cancel the common terms.

.

The function is continuous for all values of .

The removable discontinuity at .

(c) A function that have both removable discontinuity and non removable discontinuity.

Let the function is .

The denominator of the function should not be zero.

The function is discontinuous at  and .

Rewrite the function as .

Cancel the common terms.

.

The function is continuous for all values of , except .

The function has removable discontinuity at .

The function has non removable discontinuity at .

(a). Example function with a non removable discontinuity at  is .

(b). Example function with a removable discontinuity at  is .

(c). Example function that have both removable discontinuity and non removable discontinuity .