Question on the ''limit'' of Riemann integrable functions Let $x<y$ be real numbers and $f:[x,y]\to\mathbb{R}$ a bounded function. Suppose that the restriction
$$
f_{\epsilon}:[x+\epsilon,y]\to\mathbb{R}
$$
is Riemann integrable for each $\epsilon>0$. 

Does $f$ have to be Riemann integrable? If yes, does $$ \int_x^y f(x)dx=\underset{\epsilon\to 0}{\lim}\int_{x+\epsilon}^y f(x) dx $$ hold?

I think I can prove this but I am not sure that I am not proving nonsense. My ''proof'' relies much on the fact that $f$ is bounded by some $C>0$, then I thought that
$$
-C\cdot\epsilon\leq I^-_\epsilon\leq I_\epsilon^+\leq C\cdot\epsilon
$$
where where the $I^\pm_\epsilon$ are the upper and lower sums. Can I conclude that the integral exists?
 A: Let $W$ denote the set of discontinuities of $f$. Then 
$W \cap [x+\frac{1}{n}, y]$ has Lesbegue measure zero thus, $\cup_n W \cap [x+\frac{1}{n}, y]=W \cap (x,y] $ has Lesbegue measure zero thus. Hence $W$ has Lesbegue measure $0$.
Since $f$ is bounded and continuous almost everywhere it is RI..
This proof looks like an overkill though...
For the second Question, the answer is yes. Indeed:
$$\left| \int_x^y f(t)dt-\int_{x+\epsilon}^y f(t)dt \right| =\left| \int_{x+\epsilon} f(t)dt \right| \leq C(x+\epsilon-x)=C\epsilon \,.$$
A: The answer to both questions is yes.  See $\S$ 8.3.2 of these notes.
Added: I will reproduce the answer to the first question.  We show $f$ is integrable on $[a,b]$ via Darboux's Criterion: for every $\epsilon > 0$, we show there is a partition $\mathcal{P}$ of $[a,b]$ with $U(f,\mathcal{P}) - L(f,\mathcal{P}) < \epsilon$.  To do this, let $M$ be an upper bound for $|f|$ on $[a,b]$ and choose a "small" $\delta > 0$: the precise choice in terms of $\epsilon$ and $M$ will be given later.  We take $a+\delta$ as the first nonendpoint of our partition.  Then $\sup(f,[a,a+\delta]) \cdot (a+\delta-a) - 
\inf(f,[a,a+\delta]) \cdot (a+\delta -a) \leq 2M \delta$.  On $[a+\delta,b]$ $f$ is assumed to be integrable so we can find a partition $\mathcal{P}'$ such that $U(f|[a+\delta,b],\mathcal{P}') - L(f|[a+\delta,b],\mathcal{P}') < \frac{\epsilon}{2}$.
Putting $\mathcal{P} = \{a\} \cup \mathcal{P'}$ we have
$U(f,\mathcal{P}) - L(f,\mathcal{P}) < 2M \delta + \frac{\epsilon}{2}$.
Having fixed everything else first, we may choose $\delta$ so that $2M \delta < \frac{\epsilon}{2}$, and thus we have verified Darboux's Criterion.
The second statement follows almost immediately.
