Question

The total number of point of discontinuity of f(x) = [2x]² - {2x}² in the interval (-2 , 2) are :-

( The two brackets denote greatest integer and fractional part function respectively)​

Answer 1

Given

A function [2x]² - {2x}² where the two brackets denote greatest integer and fractional part function respectively.

To Find

Number of point of discontinuity of the function.

Concept

• x = [x] + {x}

• The greatest integer function and fractional part function are discontinuous at integral points.

Solutions

f(x) = [2x]² - {2x}²

=> f(x) = ( 2x - {2x} )² - {2x}²

=> f(x) = 4x² + {2x}² - 4x{2x} - {2x}²

=> f(x) = 4x² - 4x{2x}

The function will be discontinuous at function where {2x} is discontinuous. {2x} is discontinuous at integral points.

{2x} is discontinuous when 2x = integer

-2 < x < 2

=> -4 < 2x < 4

Thus, integral values of 2x is -3, -2,-1 , 0 ,1 ,2, 3.

Hence, There are total 6 points