= Finite iterated integrals without joint integrability
{title2=$f(x,y)=xy/(x^2+y^2)^3$}
Define the displayed <function> away from the origin and set $f(0,0)=0$. Every section in either coordinate is integrable on $[-1,1]$ and is an <odd function>, so both orders of <iterated integrals> on $[-1,1]^2$ are zero. However, on the annulus $\varepsilon<\sqrt{x^2+y^2}<1$, <polar integration> gives
$$
\int|f|=\int_\varepsilon^1r^{-3}\,dr\int_0^{2\pi}|\cos\theta\sin\theta|\,d\theta=\varepsilon^{-2}-1.
$$
Thus the <function> is not <Lebesgue integrable> jointly. Reflection interchanges its <positive part> and <negative part>, whose <integrals> are both infinite. Existence and even agreement of finite <iterated integrals> do not establish the <absolute integrability> hypothesis of <Fubini's theorem>.
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