Noah's proof from Tuesday is bogus. In the lexicographic ordering, if x_0 < x_1, then (x_1, y_0) > (x_0, y_n) for any y_n. What you want to do is restrict to some horizontal line, and take some decreasing sequence of points {x_n} converging to {x_0}. Pick some y_1 > y_0, and pick some epsilon less than f(x_0,y_1) - f(x_0,y_0). Since f is continuous, there must be some N for which f(x_N,y_0) - f(x_0,y_0) < epsilon < f(x_0,y_1) - f(x_0,y_0). We conclude that f(x_N,y_0) < f(x_0,y_0), which is a contradiction, since x_N > x_0.

See, i can still do math. :)