I have a question as follow:
Let $f:J\to\mathbb{R}$ be bounded with $J$ an interval with end point $a\lt b$. Show that if $f$ Riemann Integrable on all $[x,y]$, $a\lt x\leq y\lt b$ then $f$ Riemann Integrable on $J$ and $$\large\int_a^bf=\lim_{\substack{x\to a\\y\to b}}\int_x^yf.$$
May I am missing something but I have showed this (more rigorously than typed here) by considering $[a+\epsilon,b-\delta]$, finding minorant and majorant such that $I(\phi^+)-I(\phi^-)\lt \epsilon$ since it is integrable here, and extending their domain to $(a,b-\delta)$ by setting the new values to $\substack{+\\-}M$ (the bound for $f)$ respectively. Then we can say $$I(\phi_+)-I(\phi_-)\leq \epsilon +2M\epsilon$$ noting $\epsilon$ arbitrary and we can repeat this argument for $\delta$ in the opposite side of the interval to conclude.
This does not seem like a lot of work for what is supposed to be about 15mins in an exam. In my course I do need to prove integration from the Darboux definition and prove equivalence to Riemann integration in both directions, but no other mention of the Riemann integral, so it is possible that the marks for this could be coming from the time needed to work both direction, or maybe I am missing something so any hints would be great!
Hints:
(1) Given $a < \alpha < \beta < b$, there is a partition $P'$ of $[\alpha, \beta]$ where upper and lower Darboux sums satisfy $U(P',f) - L(P',f) < \epsilon/3$. Now take $\alpha$ and $\beta$ close enough to $a$ and $b$, respectively, such that $U(P,f)- L(P,f) < \epsilon$ where $P$ is the partition of $[a,b]$ obtained by adding the points $a$ and $b$ to $P'$.
Use the fact that if $M = \sup_{x\in [a,b]}|f(x)|$, then $\sup_{x\in [a,\alpha]}f(x) < M$, $\inf_{x\in [a,\alpha]}f(x) > -M$, etc.
(2) Note that if $f$ is Riemann integrable on $[a,b]$ then it is also integrable on $[a,x]$ and $[y,b]$ with
$$\left|\int_x^y f(t) \, dt - \int_a^b f(t) \, dt \right| \leqslant \int_a^x |f(t)| \, dt+ \int_y^b|f(t)|\, dt \leqslant M(x-a) + M(b-y)$$