My question like this:
Give an example of a continuous and bounded function $f: 0,2]\rightarrow R$ and a sequence of partitions ${P_n}$ such that each $P_n$ is composed of $n$ subintervals and such that $\lim_{n \rightarrow \infty}{}L(f, P_n)$ exists but does not equal $\int_a^bf$. Choose your $f$ so the integral is geometrically obvious as area. Explain why this does not contradict the AR Theorem.
I tried several different functions but always came up with the same answer by calculating either the integral $\int_0^2f$ or the $\lim_{n \rightarrow \infty}{}L(f, P_n)$. Can anyone please help with this? Thanks
Note that if $f$ is continuous on $[0,2]$ then it is integrable on $[0,2]$ and hence the limit of $L(f, P_n) $ is $\int_0^2 f$ unless the norm / mesh of $P_n$ does not tend to $0$. Thus you need to choose the right partition whose mesh does not tend to $0$ and take any non-constant function. Thus let $f(x) =x$ and let $P_n$ be given by $$P_{n} =\{0,1/(n-1), 2/(n-1),\dots,(n-1)/(n-1),2\} ,n\geq 2$$ and then you can see that the integral $\int_0^2 f=2$ and $\lim_{n\to\infty} L(f, P_n) =3/2$.
Also what do you mean by AR theorem? If you can describe it in your question, I can offer some help for that part of the question too.