True or false?
The function $$f:[0,10]\to\mathbb R\mid f(x)=x-[x]$$ is Riemann-Integrable.
Here $[x]$ is the floor function, and I think $x-[x]$ is the mantissa function:
I think the statement is true, but I don't know how to prove in a easy way. Maybe proving that if $f$ is bounded (it's true) then $$\underline{\displaystyle\int_a^bf(x)\;\text dx}\quad=\quad\overline{\displaystyle\int_a^bf(x)\;\text dx}\,,\tag 1$$ but this could be a little tedious?
What are the hypothesis, and my thesis is true? We need to check $(1)$?
Any ideas?
Thank you!
If you want to prove this by way of $$ \underline{\displaystyle\int_a^bf(x)\;\text dx}\quad=\quad\overline{\displaystyle\int_a^bf(x)\;\text dx}\,,\tag 1 $$ then it can be done. Take any $\epsilon>0$. We want to make a partition $P = (x_0, x_1, \ldots,x_n)$ of $[0,10]$ (with $x_0 = 0, x_n = 10$, and $x_i<x_{i+1}$) such that $$ U(f, P) - L(f, P)<\epsilon $$ where $U(f, P)$ is the upper sum of $f$ with respect to the partition $P$ and $L(f, P)$ is the lower. It's not too hard to show that any partition where $x_{i+1}-x_i < \epsilon/20$ will work.