Solve all your doubts with Teachoo Black (new monthly pack available now!)

Examples

Example 1

Example 2

Example 3

Example 4 Important

Example 5

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11 Important

Example 12

Example 13 Important

Example 14

Example 15 Important You are here

Example 16

Example 17 Important

Example 18

Example 19

Example 20 Important

Example 21

Example 22

Example 23 Important

Example 24

Example 25

Example 26 Important

Example 27

Example 28

Example 29 (i)

Example 29 (ii) Important

Example 29 (iii) Important

Example 29 (iv)

Example 30 Important

Example 31

Example 32 Important

Example 33 Important

Example 34

Example 35

Example 36 Important

Example 37 Important

Example 38

Example 39 Important

Example 40

Example 41 Important

Example 42 Important Deleted for CBSE Board 2023 Exams

Example 43 Deleted for CBSE Board 2023 Exams

Example 44 (i)

Example 44 (ii) Important

Example 44 (iii) Important

Example 45 (i)

Example 45 (ii) Important

Example 45 (iii) Important

Example 46

Example 47 Important

Example 48

Last updated at Aug. 19, 2021 by Teachoo

Example 15 (Introduction) Find all the points of discontinuity of the greatest integer function defined by π (π₯) = [π₯], where [π₯] denotes the greatest integer less than or equal to π₯Greatest Integer Function [x] Going by same Concept Example 15 Find all the points of discontinuity of the greatest integer function defined by π(π₯) = [π₯], where [π₯] denotes the greatest integer less than or equal to π₯Given π(π₯) = [π₯] Here, Continuity will be measured at β integer numbers, and non integer numbers Thus, we check continuity for When x is an integer When x is not an integer Case 1 : When x is not an integer f(x) = [x] Let d be any non integer point Now, f(x) is continuous at π₯ =π if (π₯π’π¦)β¬(π±βπ ) π(π)= π(π ) Value of d can be 1.2, β3.2, 0.39 (π₯π’π¦)β¬(π±βπ ) π(π) = limβ¬(xβπ) [π₯] Putting x = d = [π] π(π ) =[π] Since limβ¬(xβπ) π(π₯)= π(π) π(π₯) is continuous for all non-integer points Case 2 : When x is an integer f(x) = [x] Let c be any non integer point Now, f(x) is continuous at π₯ =π if L.H.L = R.H.L = π(π) if (π₯π’π¦)β¬(π±βπ^β ) π(π)=(π₯π’π¦)β¬(π±βπ^+ ) " " π(π)= π(π) Value of c can be 1, β3, 0 LHL at x β c limβ¬(xβπ^β ) f(x) = limβ¬(hβ0) f(c β h) = limβ¬(hβ0) [πβπ] = limβ¬(hβ0) (πβπ) = (πβπ) RHL at x β c limβ¬(xβπ^+ ) g(x) = limβ¬(hβ0) g(c + h) = limβ¬(hβ0) [π+π] = limβ¬(hβ0) π = π Since LHL β RHL β΄ f(x) is not continuous at x = c Thus, we can say that f(x) is not continuous at all integral points.