Did Fubini's Theorem Fail?

[aldrinkleys]

Did Fubini's Theorem fail here?

 

[micromass]

Fubini's theorem didn't fail here, because the conditions are not satisfied.

Roughly, Fubini's theorem (in our case) states that if $$f:[0,1]^2\rightarrow [-\infty,+\infty]$$ is a continuous function, such that one of following conditions holds:
1) $$f(x,y)\geq 0$$ forall x,y in [0,1]
2) $$\iint_{[0,1]^2}{|f(x,y)|dxdy}<+\infty$$
then
$$\iint_{[0,1]^2}{f(x,y)dxdy}=\int_0^1{\int_0^1{f(x,y)dy}dx}=\int_0^1{\int_0^1{f(x,y)dx}dy}$$

The problem is that neither (1) nor (2) are satisfied in this case.

 

[aldrinkleys]

Thank you!