For the following, let $E=[0,1]\times[0,1]$ and let $S$ be the subset of $E$ defined by $$S=\{(x,y)\in E \ | \ x=\dfrac{2j+1}{2^n}, y=\dfrac{2k+1}{2^n}, j,k=0,...,2^{n-1}-1, n\in \mathbb{N}\}$$ Let $f:E\to R$ be defined by $$f(x,y)=\begin{cases} 1, & (x,y)\in S \\ 0, & (x,y) \in E\backslash S \end{cases}$$
(a) If it exists, compute $\int_0^1 \int_0^1 f(x,y)dydx$. If it doesn't exist, explain why not.
(b) If it exists, compute $\int_0^1 \int_0^1 f(x,y)dxdy$. If it doesn't exist, explain why not.
(c) Is $f$ Riemann integrable on $E$? Why or why not?
This is what I'm stuck on. I'm thinking that if $x$ and $y$ as described are each dense in $[0,1]$, then both (a) and (b) are $1$; however, if this is the correct line of thinking, I'm unsure of how to go about showing it. Also I have a gut feeling that (c) is no, but again I'm not sure how I'd show that.