MATH 3220 § 1 FIFTH HOMEWORK ASSIGNMENT  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).
    1. 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.
    2. 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.