munkres analysis integration question Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.


Consider the partition $P$ like the above picture.  
$U(f, P) - L(f, P) = \frac{3 n - 2}{n^2}$.  
$\frac{3 n - 2}{n^2} \to 0 (n \to \infty)$.  
So, for any $\epsilon > 0$, there exists a partition $P$ such that $U(f, P) - L(f, P) < \epsilon$.  
By Theorem 10.3 (p.86), $f$ is integrable over $[0, 1] \times [0, 1]$.
