Homework 6
Assignment: Read pages 103 to 111 in Rudin. Do problems 1, 2, 6, 22. On
22, you may find it helpful to refer back to problem 6 on Homework 5.
Also do these problems:
1. Let f be a continuous function on R. Suppose f ′ (x) exists for all x ̸= 0
and f ′ (x) → 7 as x → 0. Does it follow that f ′ (0) exists?
2. Define a function f : R → R as
{ 2
x sin(1/x), x ̸= 0
f (x) =
0,
x=0
Show that f is differentiable for all x and that
{
2x sin(1/x) − cos(1/x), x ̸= 0
′
f (x) =
0,
x=0
Then show that f ′ (x) is discontinuous at x = 0. Thus the derivative
of a function may be defined in a neighborhood of a point, but not
continuous at that point.
3. Define a function f : R → R as
{
x + 2x2 sin(1/x), x ̸= 0
f (x) =
0,
x=0
(a) Show that
{
′
f (x) =
1 + 4x sin(1/x) − 2 cos(1/x), x ̸= 0
1,
x=0
(b) Show that (even though f ′ (0) > 0) f is not monotone increasing
in any neighborhood of x = 0.
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