B. Classification of Discontinuity
Definition :– Let a function f be defined in the neighbourhood of a point c, except perhaps at c itself.
Also let both one–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 ∈ R, x ≠ 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 → 0– and x → 0+, and does not tend to infinity as x → 0 There is no limit as x → 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 –1 to 1 as x→ 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 ≠ 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.
(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.