The problems are due on November 18. This is a group assignment. Turn in only one one solution per group.
Problem 1.
Let 
be a continuous function on 
so that
both
and
Show that f is uniformly continuous on .
Problem 2.
A set is called an 
-set, if it is the countable union
of closed sets.
a. Show that the interval [0,1) is an -set.
b. Show that every -set is the countable union
of compact sets.
Problem 3.
Use the previous problem to show that the continuous image
of an -set is again an -set: Let
be continuous on F, and let F be an
-set. Then f(F) is an -set.
Problem 4.
Let 
be a continuous function. Show that f has a fixed point, i.e., there is an 
satisfying f(x)=x. Hint: Consider the function f(x)-x.
Problem 5.
A function 
has the
intermediate value property, if for all 
and all 
there is an 
satisfying f(x)=y.
Let be a one-to-one function with the intermediate value property. Show that f is monotone and continuous.