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

We remind that the function $[x]$ has on the interval $[0,3]$ the following form: $[x]=0,$ $0\leq x\leq1$, $[x]=1,1\leq x\leq2$, $[x]=2,2\leq x\leq3$. It is clear that the function $f(x)=[x]-x$ is integrable on intervals $[0,1],[1,2]$ and $[2,3]$. Thus the function $x-[x]$ is integrable on the interval$[0,3]$, because it is bounded and has finitely many points of discontinuity.