Double Integrals and Iterated Integrals – Analysis

Question

Give an example (if any) for a non-integrable function $f:\mathbb{R\times R}$ $\to$
$\mathbb{R}$ with domain in $[0,1]^2$ such that both iterated integrals exists(i.e. in both order of integration).
Here is what I have got:
$$
f(x,y) =
\begin{cases}
e^{-xy}\sin x \sin y, & \text{if }x,y \geq 0 \\
0, & \text{otherwise }
\end{cases}
$$
Does this function work for my case? I found this in :http://www.mathnet.or.kr/mathnet/kms_tex/80630.pdf. It says that iterated integrals exist but not double integrals. I am not sure if this implies Riemann integral does not exist.

Answer 1

Consider the function

$$ f(x,y)= \frac{x^2-y^2}{(x^2+y^2)^2}. $$

Now, if you evaluate the integral

$$ \int_{0}^{1}\int_{0}^{1}f(x,y)dydx = \frac{\pi}{4},$$

and if you consider the other order, you get

$$ \int_{0}^{1}\int_{0}^{1}f(x,y)dxdy = -\frac{\pi}{4}. $$

So, the iterated integrals exist, but the double integral does not.