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
NCERT Exemplar - MCQs
Question 2 Important You are here
Question 3
Question 4
Question 5
Question 6 Important
Question 7 Important
Question 8
Question 9 Important
Question 10
Question 11
Question 12 Important
Question 13
Question 14 Important
Question 15
Question 16 Important
Question 17
Question 18 Important
Question 19
Question 20
Question 21 Important
Question 22
Question 1 Important
Question 2 You are here
Question 3
Question 4 Important
NCERT Exemplar - MCQs
Last updated at Sept. 21, 2024 by Teachoo
This question is similar to Example 15 - Chapter 5 Class 12 Continuity and Differentiability
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)