Question

Let g(x) = [|x|], where [·] represents greatest integer function then number of points of discontinuity of g(x), x∈ (–5, 5) is

Answer 1

Given : g(x) = [|x|], where [·] represents greatest integer function

To Find : number of points of discontinuity of g(x), x∈ (–5, 5)

Solution:

x = 0⁺

| x | = 0⁺

[0⁺]=0

x = 0⁻

| x | = 0⁺

[0⁺]=0

g(0) = 0

Hence continuous at x = 0

x = 1⁺

| x | = 1⁺

[1⁺]=1

x = 1⁻

| x | = 1⁻

[1⁻]=0

as 1  ≠ 0

Discontinuous at x = 1

similarly Discontinuous at x = 2 , 3 , 4

x = -1⁺

| x | = 1⁻

[1⁻]=0

x = -1⁻

| x | = 1⁺

[1⁺]=1

as 0  ≠ 1

Discontinuous at x = -1

similarly Discontinuous at x = -2 , -3 , -4

Hence number of points of discontinuity of g(x) = 8

-4 , - 3 , - 2 , - 1 ,  1 , 2 , 3 , 4

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Answer 2

SOLUTION

GIVEN

Let g(x) = [|x|], where [·] represents greatest integer function

TO DETERMINE

The number of points of discontinuity of g(x) , x ∈ (–5, 5)

CONCEPT TO BE IMPLEMENTED

A function f is said to be continuous at a point c if

$\displaystyle \sf{\lim_{x \to c + } \: f(x) = \lim_{x \to c - } \: f(x) = f(c)}$

EVALUATION

Here the given function is g(x) = [ | x | ]

where [·] represents greatest integer function

Let us examine the continuity of g(x) at an arbitrary number c

Then two cases arise

• c is an integer

• c is not an integer

Case : 1 : c is an integer

Let c = 2 then

$\displaystyle \sf{\lim_{x \to 2 + } \big[ \: |x| \: \big ] = 2}$

$\displaystyle \sf{\lim_{x \to 2 - } \big[ \: |x| \: \big ] = 1}$

$\displaystyle \sf{g(2) = 2}$

Thus we have

$\displaystyle \sf{\lim_{x \to 2 + } \: g(x) \ne \lim_{x \to 2 - } \: g(x)}$

So g(x) is not continuous at x = 2

Again

$\displaystyle \sf{\lim_{x \to 0 + } \: g(x) = \lim_{x \to 0 - } \: g(x) = g(0)}=0$

So g(x) is continuous at x = 0

We can conclude that g(x) is not continuous at x = c where c is an integer except 0

Consequently g(x) is not continuous at x = - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4

Case : 2 : c is not an integer

Let c = 1.5 then

$\displaystyle \sf{\lim_{x \to 1.5 + } \big[ \: |x| \: \big ] = 1}$

$\displaystyle \sf{\lim_{x \to 1.5 - } \big[ \: |x| \: \big ] = 1}$

$\displaystyle \sf{g(1.5) = 1}$

Thus we have

$\displaystyle \sf{\lim_{x \to 1.5 + } \: g(x) = \lim_{x \to 1.5 - } \: g(x) = g(1.5)}$

So g(x) is continuous at x = 1.5

We can conclude that g(x) is continuous at x = c where c is not an integer

Conclusion :

g(x) is not continuous at x = - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4

The number of discontinuous points are 8

FINAL ANSWER

Hence in the interval ( - 5 , 5) the number of points of discontinuity of g(x) are 8

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