Decide if $f:[-1,1] \rightarrow \mathbb{R}, f(x)= \mathbb{I}_{1/n}, n=1,2,3,\ldots$ is integrable using the definition of riemann sum.
So, I defined $\delta_n=1/2^{n+1}$ and $J_n=B_{\delta_n}(1/n)$ (where n=2,3,..), also, I defined $P_1=[-1,-1+\delta)\cup [1,1-\delta)\bigcup J_n$ and $P_2=P_1^c\cap [-1,1]$. Finally, P=$P_1\cup P_2$. I noted that the integral in $P_2$ is iqual 0, because f is constant. I just have to avaliate the function in $P_1$.
So, |S(f,P) - I(f,P)| =
|S(f,$P_1$) - I(f,$P_1$)| =
|$\sum{supf|_J.|J|}$ - $\sum{inff|_J.|J|}$| =
but, since $inf|_J=0$ and $supf|_J = 1$, we have
| $\sum{2\delta_n}$ | = | $\sum{1/2^n}$ | = 1
Is that correct? I'm not sure, because we know that if the set of discontinuities has null length, the function is integrable