Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

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 lim┬(x→𝑑) 𝑓(π‘₯)= 𝑓(𝑑) Value of d can be 1.2, βˆ’3.2, 0.39 lim┬(x→𝑑) 𝑓(π‘₯) = 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 lim┬(x→𝑐^βˆ’ ) 𝑓(π‘₯)=lim┬(x→𝑐^+ ) " " 𝑓(π‘₯)= 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’1) 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 Hence, f(x) is not continuous at all integral points.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.