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**B. Classification of Discontinuity **

**Definition :â€“ **Let a function f be defined in the neighbourhood of a point c, except perhaps at c itself.

Also let both oneâ€“sided limits

Then the point c is called a discontinuity of the first kind in the function f(x).

In more complicated case may not exist because

one or both one-sided limits do not exist. Such condition is called a discontinuity of the second kind.

has a discontinuity of the first kind at x = 0

The function y = |x| /x is defined for all x âˆˆ R, x â‰ 0;

but at x = 0 it has a discontinuity of the first kind.

has no limits (neither one-sided nor two-sided) at x = 2 and x = 3 since Therefore x = 2 and x = 3 are discontinuities of the second kind

The function y = ln |x| at the point x = 0 has the limits Consequently, (and also the one-sided limits) do not exist; x = 0 is a discontinuity of the second kind.

It is not true that discontinuities of the second kind only arise when

The situation is more complicated.

Thus, the function y = sin (1/x), has no one-sided limits for x â†’ 0^{â€“} and x â†’ 0^{+}, and does not tend to infinity as x â†’ 0 There is no limit as x â†’ 0 since the values of the function sin (1/x) do not approach a certain number, but repeat an infinite number of times within the interval from â€“1 to 1 as xâ†’ 0.

**Removable & Irremovable Discontinuity**

(a) In case exists but is not equal to f(c) then the function is said to have a removable discontinuity. In this case we can redefine the function such that = f(c) & make it continuous at x = c.

**Removable Type Of Discontinuity Can Be Further Classified As :**

**(i) Missing Point Discontinuity : ** exists finitely but f(a) is not defined .

e.g. has a missing point discontinuity at x = 1

**(ii) Isolated Point Discontinuity : ** exists & f(a) also exists but

x â‰ 4 & f (4) = 9 has a break at x = 4.

(b) In case does not exist then it is not possible to make the function continuous by redefining it . Such discontinuities are known as non - removable discontinuity.

**Irremovable Type Of Discontinuity Can Be Further Classified As :**

**(i) Finite discontinuity :** e.g. f(x) = x - [x] at all integral x.

**(ii) Infinite discontinuity : **

**(iii) Oscillatory discontinuity : ** e.g. f(x) = sin1/x at x = 0

In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but does not exist.

**Remark : **

**(i)** In case of finite discontinuity the non-negative difference between the value of the RHL at x = c & LHL at x = c is called **The Jump Of Discontinuity **. A function having a finite number of jumps in a given interval I is called a **Piece-wise Continuous** or **Sectionally Continuous** function in this interval.

**(ii)** All Polynomials, Trigonometrical functions, Exponential & Logarithmic functions are continuous in their domains.

**(iii) **Point functions are to be treated as discontinuous is not continuous at x = 1.

**(iv) **If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is continuous at x = c.

are continuous at x = 0 , hence the composite will also be continuous at x = 0.

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