Question

Prove that the function f(x, y) = |xy| / xy is integrable on R= [−1,1]×[−1,1].

Answer 1

Define function $f$ on $A=\{0\}\times[-1,1]\cup[-1,1]\times\{0\}$ as $f(t)=0$ for all $t\in A$ . Since $|f(t)|=1$ for all $t\in[-1,1]^2\setminus A$, we have that $|f(t)|\le 1$ on $[-1,1]^2$, that is $f$ is bounded on $[-1,1]^2$.

Next, area of $A$ equals 0, and $f$ is continuous on $[-1,1]^2\setminus A$ , because $[-1,1]^2\setminus A$ is disjoint union of $(0,1]^2$, $(0,1]\times[-1,0)$, $[-1,0)^2$ and $[-1,0)\times(0,1]$ , and $f$ is continuous on every of these sets.

Since $[-1,1]^2$ is Jordan mesurable, $f$ is bounded and continuous almost everywhere on $[-1,1]^2$, by Lebesgue's criterion for Riemann integrability we obtain that $f$ is integrable on $[-1,1]^2$ .