FIFTH HOMEWORK ASSIGNMENT MATH 3220 § 1 Due Friday, A. Treibergs September 28, 2007.

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.
• Exercises from the text "Foundations of Analysis" by Joseph L. Taylor.
• 7.5 [ 1, 2*, 3, 4, 6*, 11*, 12 ]
• 8.1 [ 1, 3*, 5* ]
• Additional exercises.  A* Prove that the "topologist's sine curve" E in the plane is connected. E = { (0,y) : -1 < y < 1} ∪ { (x, sin(1/x)) : 0 < x < 1} B* Suppose I and J are open intervals in the line and a∈I and b∈J. Suppose that f:I x J - {(a,b)} → R is a function such that for all x ∈ I-{a} the limit exists g(x) =  limy→bf(x,y) and that for all y ∈ J-{b} the limit exists: h(y) =  limx→af(x,y). Show that even though the "iterated limits" may exist L = limx→ag(x),        M = limy→bh(y), it may be the case that L ≠ M. Show, then, that the two-dimensional limit lim(x,y)→(a,b)f(x,y) fails to exist. Suppose in addition to the existence of the iterated limits one knows that the two dimensional limit exists: f(x,y)→N as (x,y)→(a,b). Show that then L = M = N.