Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE PDF Download

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

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.

1. Removable Type of Discontinuity Can Be Further Classified as

• Missing  Point  Discontinuity:  exists finitely but f(a)  is not defined.  e.g.     has  a  missing  point discontinuity at  x = 1 
• Isolated  Point  Discontinuity :   exists & f(a) also exists but  
   x ≠ 4 &  f (4) = 9 has a break  at x = 4.

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.

2. Irremovable Type Of Discontinuity Can Be Further Classified as

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