Prove the theorem: Let f(x, y) be continuous in domain D. Let f(x, y) be positive for at l...

Prove the theorem: Let f(x, y) be continuous in domain D. Let f(x, y) be positive for at least one point of D and negative for at least one point of D. Then f(x,y) = 0 for at least one point of D. [Hint: Use Problem 7 to conclude that the set A where f(x, y) > 0 and the set B where f(x, y) < 0 are open. If f(x, y) ≠ 0 in D, then D is formed of the two nonoverlapping open sets A and B; this is not possible by Problem 8.]

Remark This result extends the intermediate value theorem to functions of two variables.

Problem

Let D be a domain in the plane. Show that D cannot consist of two open sets E1, E2 with no point in common. [Hint: Suppose the contrary and choose point P in E1 and point Q in E2; join these points by a broken line in D. Regard this line as a path from P to Q and let s be distance from P along the path, so that the path is given by continuous functions x = x(s), y = y(s), 0 ≤ s ≤ L, with s = 0 at P and s = L at Q. Let f(s) = −1 if (x(s), y(s)) is in E1 and let f(s) = 1 if (x(s), y(s)) is in E2. Show that f(s) is continuous for 0 ≤ s ≤ L. Now apply the intermediate value theorem: If f(x) is continuous for a ≤ x ≤ b and f(a) ≤ 0, f(b) > 0, then f(x) = 0 for some x between a and b (see Problem 5 following Section 2.23).]