Main Problem. Let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function. If $I\subseteq [a,b]$ then prove or disprove that $f$ is Riemann integrable on $I$.
I know that if $I$ be a closed an bounded interval then we can say that $f$ is Riemann integrable. Basing upon this I tried to prove the claim but for arbitrary subsets, I can't figure out how to take the partitions. On the other hand trying to construct a counterexample seemed to be more difficult to me.
However, if the answer to the above question is "No", then my question is,
Let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function. If $I\subseteq [a,b]$ then what condition(s) on $I$ ensure that $f$ is Riemann integrable on $I$?
I tried to approach the above problem too but what I actually tried to prove was the following,
Let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function. If $I\subseteq [a,b]$ then $f$ is Riemann integrable on $I$ iff $I$ is closed and bounded.
But again I couldn't succeed. Can anyone help?