I am looking for a function $f$ defined over $Q=\left[0,1\right]\times\left[0,1\right]$, with both iterated integrals defined, but $\int_Q f$ does not exist.
I found some examples here, but I am curious if there is any general patterns underlying these functions. I remember that in differential analysis, $$\frac{\partial f(x,y)}{\partial x \partial y}=\frac{\partial f(x,y)}{\partial y \partial x}\iff\text{$f(x,y)$ is $C^2$}.$$ I am looking for similar conditions for iterated integrals to agree, or $$\int_0^1\int_0^1 f\text{ }dx\text{ }dy=\int_0^1\int_0^1 f\text{ }dy\text{ }dx\iff??$$