Example 15 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 15 (Introduction) Find all the points of discontinuity of the greatest integer function defined by 𝑓 (𝑥) = [𝑥], where [𝑥] denotes the greatest integer less than or equal to 𝑥 Greatest Integer Function [x] Going by same Concept Example 15 Find all the points of discontinuity of the greatest integer function defined by 𝑓(𝑥) = [𝑥], where [𝑥] denotes the greatest integer less than or equal to 𝑥Given 𝑓(𝑥) = [𝑥] 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 integerCase 1 : When x is not an integer f(x) = [x] Let d be any non integer point Now, f(x) is continuous at 𝑥 =𝑑 if (𝐥𝐢𝐦)┬(𝐱→𝒅) 𝒇(𝒙)= 𝒇(𝒅) L.H.S (𝐥𝐢𝐦)┬(𝐱→𝒅) 𝒇(𝒙) = lim┬(x→𝑑) [𝑥] Putting x = d = [𝑑] R.H.S 𝒇(𝒅) =[𝑑] Since lim┬(x→𝑑) 𝑓(𝑥)= 𝑓(𝑑) 𝑓(𝑥) 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 𝑥 =𝑐 if L.H.L = R.H.L = 𝑓(𝑐) if (𝐥𝐢𝐦)┬(𝐱→𝒄^− ) 𝒇(𝒙)=(𝐥𝐢𝐦)┬(𝐱→𝒄^+ ) " " 𝒇(𝒙)= 𝒇(𝒄) Value of c can be 1, −3, 0 LHL at x → c lim┬(x→𝑐^− ) f(x) = lim┬(h→0) f(c − h) = lim┬(h→0) [𝒄−𝒉] = lim┬(h→0) (𝒄−𝟏) = (𝒄−𝟏) RHL at x → c lim┬(x→𝑐^+ ) g(x) = lim┬(h→0) g(c + h) = lim┬(h→0) [𝒄+𝒉] = lim┬(h→0) 𝐜 = 𝒄 Since LHL ≠ 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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About the Author
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo