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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Last updated at April 16, 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)