We can find in any calculus textbook the following version of Fubini Theorem:
Thm. If $f:[a,b]\times [c,d]\to \mathbb{R}$ is a continuous function then we have that $f$ is Riemann integrable and $$ \iint f \,dA=\int_a^b\int_c^df(x,y)\,dy\,dx=\int_c^d\int_a^bf(x,y)\,dx\,dy. $$
My question: Is there any reasonable intuitive armument in oder to convince one that if $f$ is not a continuous function then Fubini theorem can fail?
Under certain general conditions Fubini's theorem is true for something called measurable functions (I don't know if you know this theory, possibly yes).
Look at wikipedia: https://en.wikipedia.org/wiki/Fubini%27s_theorem
This theorem holds without continuity.