Answer to Question #296226 in Real Analysis for Sarita 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

ANSWER. This is false.

EXPLANATION.

Since $[x]=\left\{\begin{matrix} 0,&0\leq x <1\\ 1,&1\leq x <2\\ 2,&2\leq x <3\\ 3,& x=3 \end{matrix}\right.$ , then $f(x) =[x]-x=\left\{\begin{matrix} -x,&0\leq x <1\\ 1-x,&1\leq x <2\\ 2-x,&2\leq x <3\\ 0,& x=3 \end{matrix}\right.$ .

For all $x\in[0,3]$ : $|f(x)|\leq 1$ . Thus $f$ is a bounded piecewise continuous function (continuous and monotonic in each interval $(k-1,k) k=1,2,3$ ). Therefore $f$ is integrable on $[0,3]$ .