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

## (A) 4Β Β Β Β

## (B) β2

## (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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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 Example 15 - 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 2 The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at (A) 4 (B) β2 (C) 1 (D) 1.5 Given π(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 π is not an integer f(x) = [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) [π₯] Putting x = d = [π] π(π ) =[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 limβ¬(xβπ^β ) f(x) = limβ¬(hβ0) f(c β h) = limβ¬(hβ0) [πβπ] = limβ¬(hβ0) (πβ1) = (πβπ) RHL at x β c limβ¬(xβπ^+ ) g(x) = limβ¬(hβ0) g(c + h) = limβ¬(hβ0) [π+π] = limβ¬(hβ0) π = π Now, we need to check at what points f(x) is continuous From options (A) 4 (B) β2 (C) 1 (D) 1.5 Only 1.5 is a non integer Hence, π(π₯) = [π₯] is continous at 1.5 So, the correct answer is (D)