 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) = lim_{y→b}f(x,y)
and that for all y ∈ J{b} the limit exists:
h(y) = lim_{x→a}f(x,y).
 Show that even though the "iterated limits" may exist
L = lim_{x→a}g(x),
M = lim_{y→b}h(y),
it may be the case that L ≠ M. Show, then, that the twodimensional 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.

