> I don't know whether or not there's a function that would give this graph, but if there is, I don't think I've learned about it yet, but anyways... Say you have a graph that looks something like a saw. It goes from say, (0,1) to (1,0) to (2,1) to (3,0) to (4,1) to (5,0) etc... You then compact the graph so much that the period was 2 points, instead of 2 units. If this could be done with some weird kind of function, it would be integrable everywhere and differentiable nowhere, right? Also, if you were to integrate from a to b, I'm guessing you'd get .5(b-a), assuming that the amplitude stayed the same.

Well, that's a good idea, and I think any example would have to be wildly oscillating over arbitrarily small intervals. But it's not quite like you described it, because that sort of function wouldn't be integrable. Here's why:

To be integrable, for any sequence of partitions of (a,b) and corresponding Riemann sums, the sums have to converge to a limit. If the function oscillated between 0 and 1 on arbitrarily small intervals, then you for each partition you could take all your sample points to be 0, or take all your sample points to be 1. That means you could obtain a sequence of Riemann sums: 0, 1, 0, 1, 0, 1, ..., even when the partitions get finer and finer. The sequence 0, 1, 0, 1, ... doesn't converge, so the function isn't integrable.

So far we've been talking about Riemann integration, which is what you use when dealing with continuous/piece-wise continuous/slightly more involved functions. However, you can use an idea idea called Lebesgue integration to deal with a slightly extended class called (surprisingly enough) Lebesgue measurable functions. An even more general idea is defining integration in terms of an arbitrary "measure" on a space. Lebesgue measure extends the usual notion of length in the real number line, i.e. the Lebesgue measure of the interval (a,b) is b-a.

Anything that's Riemann integrable is also Lebesgue integrable, and the integrals are the same. But there are example of functions which are only Lebesgue integrable, and your example reminded me of one:

On the interval (0,1), define f(x) = 0 if x is irrational, and f(x) = 1 if x is rational. It's not a Riemann integrable function for exactly the same reason as I said before: you can get a sequence of Riemann sums 0, 1, 0, 1, ... by alternatively choosing your sample points to be irrational or rational. However, it's Lebesgue integrable because there are only a countable number of points where it's non-zero, and so in fact it has a Lebesgue integral of zero.

You hypothesized that you could make a function which is 0 half the time and 1 half the time, and would have an integral of 0.5 (b-a). I don't see a way to precisely define a function like that without ruining its Lebesgue integrability.