how ro prove f(x,y) is integrable in $[a,b]\times[c,d]$ If there exits a $f(x,y)$ in $\mathbb{R}^2$,and if we fix any $x$ in $[a,b]$, then $f(x,y)$ is increasing as $y$ increases. Also, if we fix any $y$ in $[c,d]$,the $f(x,y)$ is  increasing as $x$ increases. Thus, How to prove the function is integrable?
The most important thing is that how to find a suitable partition, but I am stuck here! Or, can we use measure $0$ to prove it?
 A: Yes, I think that proving:
(a) $f$ is bounded 
and
(b) the set of points of discontinuity of $f$ has measure zero
would be easier and is equivalent to establishing the integrability of $f$. I think (a) is relatively straightforward. For (b), here is an exercise:
Exercise 1: Prove that the set of points of discontinuity of a monotonically increasing function $g:[a,b]\to \mathbb{R}$ is countable.
Can you now deduce (b) from Exercise 1?
Hope this helps!
A: Assume $f(0,0)=0$, $\>f(1,1)=1$, and partition $[0,1]^2$ into $N^4$ squares $Q_k$ of equal area ${1\over N^4}$. Denote the lower left and the upper right vertices of $Q_k$ by $\lfloor Q_k$ and $Q_k\rceil$.
A  $Q_k$ is bad if $f\bigl(Q_k\rceil\bigr)-f\bigl(\lfloor Q_k\bigr)\geq {1\over N}$, and is good otherwise. 
The $Q_k$s are  diagonally split by the $45^\circ$ ascending lines $\ell_j$ intersecting the $x$-axis at $({j\over N^2},0)$, where $|j|\leq N^2-1$. There are $2N^2-1$ such lines, and each of them can split at most $N$ bad $Q_k$s.
Let $U(f)$ and $L(f)$ be the upper and the lower Riemann sum of $f$ belonging to this partition. Then we obtain
$$U(f)-L(f)\leq \sum_{{\rm bad}\ k}1\>{1\over N^4}+\sum_{{\rm good}\ k}{1\over N}\>{1\over N^4}\leq{(2N^2-1) N\over N^4}+{N^4\over N\>N^4}<{3\over N}\ .$$
It follows that $U(f)-L(f)$ can be made arbitrarily small; whence $f$ is Riemann integrable over $[0,1]^2$.
