Riemann integrable function with floor function and characteristic function.

Question

Hellow, I have a problem with this exercise:

For nonzero real constants $c_1,c_2,\dots,c_n$. Let $f(x)=\sum_{j=1}^n c_j\lfloor x\rfloor \chi_{[j,j+1)}(x)$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function and $\chi$ denotes the characteristic function on $\mathbb{R}$. Is $f$ Riemann integrable on $\mathbb{R}$? Carefully justify the position taken; if yes, find the value of integral.

I don't understand the question because I just know the definition of riemann integral function in a interval $[a,b]$. What is the definition of Riemann integral function in $\mathbb{R}$?

Thanks.

Answer 1

Let $\varepsilon>0$ be given. Set $D=\max\{1,|2c_2-c_1|, |3c_3-2c_2|,\ldots, |nc_n-(n-1)c_{n-1}|\}$, and select $\delta = \min \left\{1, \frac{\varepsilon}{Dn} \right\}$. Now we may notice that for every subdivision $P$ of the interval $[1, n+1]$ with mesh less than $\delta$ we have \begin{aligned} U(P,f)-L(P,f)&<D(n-1)\delta \\& < \varepsilon. \end{aligned} Hence we have shown that $\underline {\int_1^{n+1}} f \left({x}\right) d x = \overline {\int_1^{n+1}} f \left({x}\right) d x$, and so $f$ is Darboux integrable over $[1, n+1]$.