Answer to Question #299948 in Real Analysis for Dhruv bartwal

Question #299948

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


True or false with full explanation

1
Expert's answer
2022-02-21T16:03:29-0500

"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 mearuse is 0. By Lebesque's criterion of integrability, the function f(x) is integrable.


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