Hello there,
how would I approach this problem?
Suppose f be a continuous function on a compact domain D. Show that f has a maximum and a minimum on D.
compact can be translated as closed and bounded, which means that f has to be a bounded function (you can prove this by contradiction, here is where you use the fact f is continuous). Since f is bounded, the set $\displaystyle \{ f(x) \mid x \in D \}$ has a supremum and infimum. Which are the maximum and minimum values of f respectively.