Let $\displaystyle f$ be bounded on $\displaystyle [a,b]$. Show that if the set of discontinuous points of $\displaystyle f$ has content zero, then $\displaystyle f$ is integrable.

I am trying do a second semester analysis self study and I am pretty stuck on a few major concepts of integration. I would appriciate some help with this one. If any thing, I just need a place to get started.

Here is what I have set up so far. I am given:

$\displaystyle D=\{d_{1}, d_{2},...,d_{N}\}$ has content zero and $\displaystyle f$ is bounded on $\displaystyle [a,b]$. Thus, $\displaystyle D\subseteq\bigcup_{n=1}^{N}O_{n}$ and $\displaystyle \sum_{n=1}^{N}|O_{n}|\leq\epsilon$ (where, $\displaystyle O_{n}$ represents open sets). Moreover, there exists an $\displaystyle M>0$ satistying $\displaystyle |f(x)|\leq\\M$ for all $\displaystyle x\in[a,b]$.

My goal is to show there exists a partition $\displaystyle P_{\epsilopn}$ such that $\displaystyle U(f,P_{\epsilopn})-L(f,P_{\epsilopn})<\epsilon$ for any choice of

$\displaystyle \epsilon>0$.

Since the goal begins with a universal quantifier I let $\displaystyle \epsilon>0$ be arbitrary. Now, I need to show the existance of such a partition.

As you can see I do not have much. I am not sure how to get started.

Q1: Does the set of open intervals whos union contain the set D have to be nested?

Q2: Can I assume $\displaystyle [a,b]$ is compact to create a finite subcover and use that somehow?

Thanks