Question 3
The number of points at which the function f (x) = 1/(𝑥−[𝑥] ) is not continuous is
(A) 1 (B) 2
(C) 3 (D) none of these
Given f(x) = 1/(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) = 1/(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) 1/(𝑥 − [𝑥] )
Putting x = d
=1/(𝑑 − [𝑑] )
𝒇(𝒅)
=1/(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) 𝟏/((𝑐 − ℎ) − [𝒄 − 𝒉])
= lim┬(h→0) 𝟏/((𝑐 − ℎ) − (𝒄 − 𝟏))
= lim┬(h→0) 𝟏/(𝑐 − ℎ − 𝑐 + 1)
= lim┬(h→0) 𝟏/(−ℎ + 1)
= 𝟏/(0 + 1)
= 𝟏/𝟏
= 1
RHL at x → c
(𝒍𝒊𝒎)┬(𝐱→𝒄^+ ) f (x) = (𝒍𝒊𝒎)┬(𝐡→𝟎) f (c + h)
= lim┬(h→0) (𝑐+ℎ)−[𝒄+𝒉]
= lim┬(h→0) (𝑐−ℎ)−(𝒄)
= lim┬(h→0) −ℎ
= 𝟎
Since LHL ≠ RHL
∴ f(x) is not continuous at x = c
Hence, f(x) is not continuous at all integral points.
∴ There are infinite number of points where f(x) = 1/(𝑥−[𝑥] ) 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)

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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