Answer to Question #301452 in Real Analysis for Dhruv bartwal

Question

The function f(x)= [x]- x is not integrable in [ 0,3] , where [ x] denote greatest integer function.

True or false with full explanation

Answer 1

Solution:

$f(x)=x-x=0,$ if $x\in [0,1).$

$f(x)=(x-1)-x=-1,$ if $x\in [1,2).$

$f(x)=(x-2)-x=-2,$ if $x\in [2,3).$

Therefore, the set of discontinuity of the function f(x) is $\{1,2,3\}$. This set is finite, hence, its measure is 0. By Lebesque's criterion of integrability, the function f(x) is integrable.