Let $D$ be a bounded, open subset of $\mathbb{R}^2$ and let $f: D ⟶ \mathbb{R}$ be a bounded, continuous function. Then is the integral $\iint _ {D}f(x,y)dA$ necessarily defined?
I tried to approach this problem by studying the behavior of the bounds. My intuition says that since $D$ is a bounded, continuous function the limits exist and hence the integral is defined. However, I'm not sure how to rigorously prove this.
Any guidance is greatly appreciated!