**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.

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

Consider the function f(x) = tanx

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.