The problems are due on October 23. This is a group assignment. Turn in only one one solution per group.
Problem 1.
Let 
be the function defined by
Show that
exists. Show that for 
,
does not exist.
Problem 2.
Let 
be a bounded function. Define 
by 
. Show that 
is a decreasing function. Then show that has
a limit at 
, if f itself has a limit at 
and
Problem 3. p. 81, # 26.
The last two problems require the following concepts:
Let 
be a function, and let be an accumulation point of
D.
Let 
be the restriction of f to
, i.e. r(x):=f(x) for all 
with
. We say
that f has a right-hand limit at (denoted by
if
exists.
Similarly we define 
to be the restriction of
f to 
. Then f is said to have a left-hand
limit at (denoted by
if
exists.
Problem 4.
Let , and let be an accumulation point of D.
Show that
exists if and only if both
exist and are equal.
Problem 5. a. Give an example of a function for which the right-hand limit at a point does not exist.
b. Show that for an increasing function 
the left-hand limit exists for all 
.