Continuity at a Point 5

Question

Give an example of a function with both a removable and a non-removable discontinuity.

Answer 1

Classification of discontinuity :

All discontinuity points are divided into discontinuities of the first and second kind.

The function f(x) has a discontinuity of the first kind at x = a if

• There exist left - hand limit  and right hand limit.
• These one - sided limits are finite.
• The right - hand limit and left hand limit are equal to each other, such a point is called a removable discontinuity.
• The right - hand limit and left hand limit are unequal : In this case the function f(x) has a jump discontinuity.
• The function f(x) is said to have a nonremovable discontinuity, at x = a, if at least one of the one sided limits either does not exist or is infinite.

Example :

The function f (x) = .

Continuity at point 5:

(i). .

(ii). .

 (iii). .

Removable discontinuity:

So, the three conditions are satisfied and the function is continues at the point 5.

Consider the function f(x) = . Then f(x) = (x + 1) for all real numbers except x = 1.

Since f(x) and (x + 1) agree at all points, other than objective,

.

We can " remove " the discontinuity by filling the hole.

The domain of f(x) may be extended to include x = 1 by declaring that f(1) = 2.

This makes f(x) continuous at x = 1.

Since, f(x) is continuous at all other points, defining f(x) = 2 turns f into a continuous function.

Nonremovable discontinuity:

The function is .

The given function is not defined at x = − 1 and x = 1. Hence, this function has discontinuities at x = ± 1. To determine

the type of the discontinuities, we find the one - sided limits :

,

.

Since the left - side limit at x = − 1 is infinity, we have an essential discontinuity at this point.

 ,

.

Similarly, the right-side limit at x = 1 is infinity. Hence, here we also have an nonremovable discontinuity.