Let $$f: D \subset \mathbf{R}^{2} \to \mathbf{R} $$ a continuous function and $ D $ a bounded and closed domain. Prove that f is integrable.
I have been reviewing my exercises and this is one I cannot get a grip on. I am trying to enforce uniform continuity by taking $$|(x,y) - (x_{0},y_{0})| < \epsilon \\ \iff |f(x,y) - f(x_{0},y_{0})| < \delta$$
But I have no clue as to how to continue. For functions of one dimension, the next step is to assume a partition $(I_{n})$ a finite partition of $[a,b]$ such that its diameter is less than $\delta$ and so on...
I have no idea how to transform that in two dimensions using squares. Any help would be greatly appreciated.