Types of discontinuity:

Basically there are two kind of discontinuity.

Removable discontinuity:

If f(x) exists but is not same to f(a), then f(x) has a removable discontinuity at x = a and it may be replaced by redefining f(x) for x = a.

Example: Redefine the function f(x) =[sinx] where x ∈(0,Π) in such a type that it  should become continuous for x ∈(0,Π).

Solution:             Here = 0  but f(Π/2).                 

      Consequently, f(x) has a removable discontinuity at x = Π/2  .

      To replace this we redefine f(x) as follows

      f(x) = [sinx],          x ∈ (0,Π/2) ∪ (Π/2,Π)

        = 0 ,                    x = Π/2.

      Now, f(x) is continuous for x ∈(0,Π).

Non-removable discontinuity:

If f(x) does not exist, then we may not replace that discontinuity. So that becomes a non- removable discontinuity or required discontinuity.

Example: Suppose that  f (x) = {x} has non removable discontinuity at any x∈I.

Solution: Since f(x) does not exist for any a ∈ I.

                Therefore, f(x)= {x} has non-removable discontinuity at any x ∈ I

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