1. ## content zero

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

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

My goal is to show there exists a partition $P_{\epsilopn}$ such that $U(f,P_{\epsilopn})-L(f,P_{\epsilopn})<\epsilon$ for any choice of
$\epsilon>0$.

Since the goal begins with a universal quantifier I let $\epsilon>0$ 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 $[a,b]$ is compact to create a finite subcover and use that somehow?

Thanks

2. If I am reading this correctly the set D has only a finite number of elements in it. So try a easier problem first. What if f is bounded and is only discontinuous at one point c in [a,b]. How could you create a partition that that would show f is integrable. If you can solve this problem it will generalize to any finite set of points of discontinuity.

Hint: squeeze the one point of discontinuity into a set of length $\frac{\epsilon}{2M}$ then the function is continuous on $[a,c-\frac{\epsilon}{4M}]$ and $[c+\frac{\epsilon}{4M},b]$

3. The concept of content is a bit dated.
If you are using the text material by T.H. Hildebrand then a subset $E\subseteq [a,b]$ has content if and only if its characteristic function $\chi_E$ is Riemann integrable on $[a,b]$.
That seems to be consistent with your notation. If so how can you use that?

If that is not the definition of content you have, what textbook are you following?

4. Originally Posted by TheEmptySet
If I am reading this correctly the set D has only a finite number of elements in it. So try a easier problem first. What if f is bounded and is only discontinuous at one point c in [a,b]. How could you create a partition that that would show f is integrable. If you can solve this problem it will generalize to any finite set of points of discontinuity.

Hint: squeeze the one point of discontinuity into a set of length $\frac{\epsilon}{2M}$ then the function is continuous on $[a,c-\frac{\epsilon}{4M}]$ and $[c+\frac{\epsilon}{4M},b]$
Well, I would want to construct a partition that minimizes the effect of the discontinuity at $c$ by embedding $c$ in a very small subinterval.

Originally Posted by Plato
The concept of content is a bit dated.
If you are using the text material by T.H. Hildebrand then a subset $E\subseteq [a,b]$ has content if and only if its characteristic function $\chi_E$ is Riemann integrable on $[a,b]$.
That seems to be consistent with your notation. If so how can you use that?

If that is not the definition of content you have, what textbook are you following?
I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.

5. Originally Posted by Danneedshelp
Well, I would want to construct a partition that minimizes the effect of the discontinuity at $c$ by embedding $c$ in a very small subinterval.

I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.
I do not know Abbots' text book.
I your have reasonable university library access you may want to find Hildebrandt's book or a book by S K Breberian.

6. A set $X\subset \mathbb{R}$ has content zero iff for every $\varepsilon >0$ there exist $(I_k)_{k=1}^{n}$ intervals with $X \subset \cup I_k$ and $\sum m(I_k) < \varepsilon$.

So in your problem, given $\varepsilon >0$ just use an argument completely analogous to that where you have a finite number of discontinuities.