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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  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEEClassification of Discontinuity | Mathematics (Maths) Class 12 - JEE
  • Then the point c is called a discontinuity of the first kind in the function f(x).
  • In more complicated case   Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE may not exist because one or both one-sided limits do not exist. Such condition is called a discontinuity of the second kind. 

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

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

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

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.
  • Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE
  • Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE  has no limits (neither one-sided nor  two-sided) at x = 2 and x = 3 since  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE  Therefore x = 2 and x = 3 are discontinuities of the second kind 

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

  • The function y = ln |x| at the point x = 0 has the limits  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE Consequently,   Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE (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 Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE 
    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.

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

 

Removable & Irremovable Discontinuity

In case  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE  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 Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE = f(c) & make it continuous at x = c.

1. Removable Type of Discontinuity Can Be Further Classified as

  • Missing  Point  Discontinuity: Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE exists finitely but f(a)  is not defined.  e.g.  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE   has  a  missing  point discontinuity at  x = 1 
  • Isolated  Point  Discontinuity :  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE exists & f(a) also exists but   Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE
    Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE x ≠ 4 &  f (4) = 9 has a break  at x = 4.

In  case   Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE 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 :  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE
  • 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  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE 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   Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE 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.  

Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE  are continuous at x = 0 , hence the  composite  Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE  will  also  be  continuous  at x = 0.

The document Classification of Discontinuity | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Classification of Discontinuity - Mathematics (Maths) Class 12 - JEE

1. What is a removable discontinuity?
Ans. A removable discontinuity, also known as a removable singularity, is a type of discontinuity in a function where a point is missing from the graph but can be filled in or defined at that point without changing the value of the function.
2. What is an irremovable discontinuity?
Ans. An irremovable discontinuity, also known as an essential singularity, is a type of discontinuity in a function where a point is missing from the graph and cannot be filled in or defined at that point without changing the value of the function.
3. How can you classify discontinuities?
Ans. Discontinuities can be classified into three main types: removable, jump, and infinite discontinuities. Removable discontinuities can be filled in or defined without changing the value of the function, jump discontinuities occur when the function "jumps" from one value to another at a certain point, and infinite discontinuities happen when the function approaches positive or negative infinity at a certain point.
4. Can a function have both removable and irremovable discontinuities?
Ans. Yes, a function can have both removable and irremovable discontinuities. Removable discontinuities are typically holes in the graph that can be filled in, while irremovable discontinuities are points that cannot be defined or filled in without changing the value of the function.
5. How do discontinuities affect the behavior of a function?
Ans. Discontinuities can significantly impact the behavior of a function. Removable discontinuities can be filled in to make the function continuous, while irremovable discontinuities indicate a fundamental change in the function's behavior at that point. Discontinuities can cause the function to have jumps, breaks, or infinite values, affecting its overall continuity and smoothness.
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