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

(A) 4     

(B) −2

(C) 1      

(D) 1.5

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

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

Transcript

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.