Check sibling questions

The number of points at which the function f (x) = 1/(x-[x] ) is not continuous is

(A) 1     

(B) 2

(C) 3      

(D) none of these

This question is similar to Ex 5.1, 19 - Chapter 5 Class 12 - Continuity and Differentiability

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Transcript

Question 3 The number of points at which the function f (x) = 1/(π‘₯βˆ’[π‘₯] ) is not continuous is (A) 1 (B) 2 (C) 3 (D) none of these Given f(x) = 1/(π‘₯ βˆ’ [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) = 1/(π‘₯ βˆ’ [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) 1/(π‘₯ βˆ’ [π‘₯] ) Putting x = d =1/(𝑑 βˆ’ [𝑑] ) 𝒇(𝒅) =1/(𝑑 βˆ’ [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 (π’π’Šπ’Ž)┬(𝐱→𝒄^βˆ’ ) f (x) = (π’π’Šπ’Ž)┬(π‘β†’πŸŽ) f (c βˆ’ h) = lim┬(hβ†’0) 𝟏/((𝑐 βˆ’ β„Ž) βˆ’ [𝒄 βˆ’ 𝒉]) = lim┬(hβ†’0) 𝟏/((𝑐 βˆ’ β„Ž) βˆ’ (𝒄 βˆ’ 𝟏)) = lim┬(hβ†’0) 𝟏/(𝑐 βˆ’ β„Ž βˆ’ 𝑐 + 1) = lim┬(hβ†’0) 𝟏/(βˆ’β„Ž + 1) = 𝟏/(0 + 1) = 𝟏/𝟏 = 1 RHL at x β†’ c (π’π’Šπ’Ž)┬(𝐱→𝒄^+ ) f (x) = (π’π’Šπ’Ž)┬(π‘β†’πŸŽ) f (c + h) = lim┬(hβ†’0) (𝑐+β„Ž)βˆ’[𝒄+𝒉] = lim┬(hβ†’0) (π‘βˆ’β„Ž)βˆ’(𝒄) = lim┬(hβ†’0) βˆ’β„Ž = 𝟎 Since LHL β‰  RHL ∴ f(x) is not continuous at x = c Hence, f(x) is not continuous at all integral points. ∴ There are infinite number of points where f(x) = 1/(π‘₯βˆ’[π‘₯] ) is not continuous Since we need to find points where f(x) is not continuous And, our options are (A) 1 (B) 2 (C) 3 (D) none of these 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.