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

## (A) 1Β Β Β Β

## (B) 2

## (C) 3Β Β Β Β Β

## (D) none of these

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Ex 5.1, 19 - Chapter 5 Class 12- Continuity and Differentiability

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Chapter 5 Class 12 Continuity and Differentiability (Term 1)

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Last updated at Nov. 17, 2021 by Teachoo

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This question is similar to

Ex 5.1, 19 - Chapter 5 Class 12- Continuity and Differentiability

Hello! Teachoo has made this answer with days (even weeks!) worth of effort and love. If Teachoo has been of any help to you in your Board exam preparation, then please support us by clicking on this link to make a donation

Question 3 The number of points at which the function f (x) = 1/(π₯β[π₯] ) is not continuous is (A) 1 (B) 2 (C) 3 (D) none of these Given f(x) = 1/(π₯ β [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 π is not an integer f(x) = 1/(π₯ β [π₯] ) Let d be any non integer point Now, f(x) is continuous at π₯=π if (π₯π’π¦)β¬(π±βπ ) π(π)=π(π ) (π₯π’π¦)β¬(π±βπ ) π(π) = limβ¬(xβπ) 1/(π₯ β [π₯] ) Putting x = d =1/(π β [π] ) π(π ) =1/(π β [π] ) Since limβ¬(xβπ) π(π₯)= π(π) β΄ π(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₯ =π if L.H.L = R.H.L = π(π) if (π₯π’π¦)β¬(π±βπ^β ) π(π)=(π₯π’π¦)β¬(π±βπ^+ ) " " π(π)= π(π) LHL at x β c (πππ)β¬(π±βπ^β ) f (x) = (πππ)β¬(π‘βπ) f (c β h) = limβ¬(hβ0) π/((π β β) β [π β π]) = limβ¬(hβ0) π/((π β β) β (π β π)) = limβ¬(hβ0) π/(π β β β π + 1) = limβ¬(hβ0) π/(ββ + 1) = π/(0 + 1) = π/π = 1 RHL at x β c (πππ)β¬(π±βπ^+ ) f (x) = (πππ)β¬(π‘βπ) f (c + h) = limβ¬(hβ0) (π+β)β[π+π] = limβ¬(hβ0) (πββ)β(π) = limβ¬(hβ0) ββ = π Since LHL β RHL β΄ f(x) is not continuous at x = c Hence, f(x) is not continuous at all integral points. β΄ There are infinite number of points where f(x) = 1/(π₯β[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)