Non-Removable Discontinuity

Non-Removable Discontinuity

If \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)\ne \underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)\), then f(x) is said to have the first kind of non-removable discontinuity.

Consider the function f(x) = 1/x. the function is not defined at x = 0. It cannot be extended to a continuous function whose domain is R. since no matter what value is assigned at 0, the resulting function will not be continuous. A point in the domain that cannot be filled in so that the resulting function is continuous is called a Non-Removable Discontinuity.

Graphical view of Non-Removable Discontinuity:

Both the limits are finite and equal: Consider the function f(x) = [x] – greatest integer function. As shown in the graph has jump of discontinuity at all integral value of x.

Graphical view of Non-Removable Discontinuity

At least one of left and right limit is infinity or vertical asymptote

Consider the function f(x) = tanx

Graphical view of Non-Removable Discontinuity 1

Here, the function is not defined at point \(\pm 3\frac{\pi }{2}\) and near these points, the function becomes both arbitrarily large and small. since the function is not defined at these points, it cannot be continuous.

Example: Find the points of discontinuity of the \(f(x)=\frac{1}{2\sin x-1}\).

Solution: Given that \(f(x)=\frac{1}{2\sin x-1}\).

f(x) is discontinuous when

2sinx – 1 = 0

Sinx = ½

x = sin⁻¹ (½)

x = π/6 (or) 5π/6

The general solution is \(2n\pi \pm \frac{\pi }{6}\) (or) \(2n\pi \pm \frac{5\pi }{6}\), n ϵ z.