Consider the function $f:[0,1]\times [0,1]\to \mathbb{R}$ $f(x,y)=1-\frac{1}{q}\space \text{if} \space x=p/q \space \text{where} \space p,q\in \mathbb{N} \text{are relatively prime and y is rational}$
otherwise $f(x,y)=1 $
a) Show that $f$ is integrable
b) show that the iterated integrals may not exist
The part a) was similar to this and adapting a similar method as coffeemath did we can show $$U(f,P)-L(f,P)<\epsilon$$ for some partition P
For the second part $\int_0^1 f(x,y)dy=f(x,y)$ so the integral clearly exists so by the fubini theorem $$\int_0^1 \left(\int_0^1 f(x,y)dy\right)dx=\int_0^1 f(x,y)dx$$ the above exists .This would then mean that $\int_0^1 f(x,y)dx$ exists and thus again the other iterated integral $\int_0^1 \left(\int_0^1 f(x,y)dx\right)dx$ exists contradicting what we were asked to prove .Where am I wrong
