I don't understand why at the beginning Spivak talks about closed rectangles whereas later it seems he implicitly refers to open rectangles (because compactness is a statenemt about open sets).
The oscillation at a point is just the limit of the oscillation in an $\epsilon$-neighborhood of that point as $\epsilon \rightarrow 0$. So I find it obvious that the first statement is true for open rectangles. But Spivak probably doesn't refer to closed ones for no reason, does he? I can't see that reason.