Question 2
The function f (x) = [x], where [x] denotes the greatest integer function, is continuous at
(A) 4 (B) −2
(C) 1 (D) 1.5
Given
𝑓(𝑥) = [𝑥]
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) = [𝑥]
Let d be any non integer point
Now, f(x) is continuous at 𝑥 =𝑑
if (𝐥𝐢𝐦)┬(𝐱→𝒅) 𝒇(𝒙)= 𝒇(𝒅)
(𝐥𝐢𝐦)┬(𝐱→𝒅) 𝒇(𝒙)
= lim┬(x→𝑑) [𝑥]
Putting x = d
= [𝑑]
𝒇(𝒅)
=[𝑑]
Since lim┬(x→𝑑) 𝑓(𝑥)= 𝑓(𝑑)
∴ 𝑓(𝑥) 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 𝑥 =𝑐
if L.H.L = R.H.L = 𝑓(𝑐)
if (𝐥𝐢𝐦)┬(𝐱→𝒄^− ) 𝒇(𝒙)=(𝐥𝐢𝐦)┬(𝐱→𝒄^+ ) " " 𝒇(𝒙)=𝒇(𝒄)
LHL at x → c
lim┬(x→𝑐^− ) f(x) = lim┬(h→0) f(c − h)
= lim┬(h→0) [𝒄−𝒉]
= lim┬(h→0) (𝑐−1)
= (𝒄−𝟏)
RHL at x → c
lim┬(x→𝑐^+ ) g(x) = lim┬(h→0) g(c + h)
= lim┬(h→0) [𝒄+𝒉]
= lim┬(h→0) 𝐜
= 𝒄
Now, we need to check at what points f(x) is continuous
From options
(A) 4 (B) −2 (C) 1 (D) 1.5
Only 1.5 is a non integer
Hence, 𝑓(𝑥) = [𝑥] is continous at 1.5
So, the correct answer is (D)

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.

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