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Example 15 - Find all points of discontinuity of greatest integer

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Example 15 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

Example 15 - Chapter 5 Class 12 Continuity and Differentiability - Part 3
Example 15 - Chapter 5 Class 12 Continuity and Differentiability - Part 4
Example 15 - Chapter 5 Class 12 Continuity and Differentiability - Part 5

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Example 15 (Introduction) Find all the points of discontinuity of the greatest integer function defined by 𝑓 (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 𝑓(π‘₯) = [π‘₯], where [π‘₯] denotes the greatest integer less than or equal to π‘₯Given 𝑓(π‘₯) = [π‘₯] Here, Continuity will be measured at – 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‘₯ =𝑑 if (π₯𝐒𝐦)┬(𝐱→𝒅) 𝒇(𝒙)= 𝒇(𝒅) Value of d can be 1.2, βˆ’3.2, 0.39 (π₯𝐒𝐦)┬(𝐱→𝒅) 𝒇(𝒙) = lim┬(x→𝑑) [π‘₯] Putting x = d = [𝑑] 𝒇(𝒅) =[𝑑] Since lim┬(x→𝑑) 𝑓(π‘₯)= 𝑓(𝑑) 𝑓(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‘₯ =𝑐 if L.H.L = R.H.L = 𝑓(𝑐) if (π₯𝐒𝐦)┬(𝐱→𝒄^βˆ’ ) 𝒇(𝒙)=(π₯𝐒𝐦)┬(𝐱→𝒄^+ ) " " 𝒇(𝒙)= 𝒇(𝒄) Value of c can be 1, βˆ’3, 0 LHL at x β†’ c lim┬(x→𝑐^βˆ’ ) f(x) = lim┬(hβ†’0) f(c βˆ’ h) = lim┬(hβ†’0) [π’„βˆ’π’‰] = lim┬(hβ†’0) (π’„βˆ’πŸ) = (π’„βˆ’πŸ) RHL at x β†’ c lim┬(x→𝑐^+ ) g(x) = lim┬(hβ†’0) g(c + h) = lim┬(hβ†’0) [𝒄+𝒉] = lim┬(hβ†’0) 𝐜 = 𝒄 Since LHL β‰  RHL ∴ f(x) is not continuous at x = c Thus, we can say that f(x) is not continuous at all integral points.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.