Riemann integrabilty of an indicator function [closed]

Take a partition $P$ of $[0,1]$ that is defined as: $$ P = \left\{0,a_1-\frac{\epsilon}{2},a_1+\frac{\epsilon}{2},\dots,a_n-\frac{\epsilon}{2},a_n+\frac{\epsilon}{2},1\right\}. $$ Clearly, $U(P,f) = \epsilon N$ and $L(P,f) = 0$. Since for any given $\epsilon>0$, a partition that could yield $$ U(P,f) - L(P,f) < \epsilon $$ can be found, $f$ is Riemann integrable.

To compute $\int f$, note that $\int f = \underline{\int f} = \sup_{P}L(P,f) = 0$, since for any given partition $P$, $L(P,f) = 0$.

HINT

You can bound the Riemann sum $I(P,\{t^*_i\})$ by $$0 \le I(P,\{t^*_i\}) \le n|P| $$ where $|P|$ is the length of the longest interval in the partition.