Check sibling questions

The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at

(A) 4Β  Β Β Β 

(B) βˆ’2

(C) 1Β  Β Β Β Β 

(D) 1.5

This question is similar to Example 15 - Chapter 5 Class 12 Continuity and Differentiability

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Question 2 The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at (A) 4 (B) βˆ’2 (C) 1 (D) 1.5 Given 𝑓(π‘₯) = [π‘₯] Since Greatest Integer Function changes value on integer numbers Thus, we check continuity When x is not an integer When x is an integer Case 1 : When 𝒙 is not an integer f(x) = [π‘₯] Let d be any non integer point Now, f(x) is continuous at π‘₯ =𝑑 if (π₯𝐒𝐦)┬(𝐱→𝒅) 𝒇(𝒙)= 𝒇(𝒅) (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 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) 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’1) = (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 Now, we need to check at what points f(x) is continuous From options (A) 4 (B) βˆ’2 (C) 1 (D) 1.5 Only 1.5 is a non integer Hence, 𝑓(π‘₯) = [π‘₯] is continous at 1.5 So, the correct answer is (D)

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.