Baby Rudin Problems
Last update: 19 March 2014
Suppose is a real function defined on which satisfies
for every Does this imply that is continuous?
If is a continuous mapping of a metric space into a metric space
for every set
denotes the closure of
Show, by an example, that
can be a proper subset of
Let be a continuous real function on a metric space Let
(the zero set of
be the set of all at which
Let and be continuous mappings of a metric space into a metric space
and let be a dense subset of Prove
that is dense in
for all prove that
for all (In other words, a continuous mapping is determined by its values
on a dense subset of its domain.)
If is a real continuous function defined on a closed set
prove that there exist continuous real functions on such that
(Such functions are called continuous extensions of
from to Show that the result becomes false
if the word "closed" is omitted. Extend the result to vector-valued functions. Hint: Let the graph of be a straight line on each of the
segments which constitute the complement of (compare Exercise 29, Chap. 2). The result remains true if
is replaced by any metric space, but the proof is not so simple.
If is defined on the graph of is the set of points
In particular, if is a set of real numbers, and
is real-valued, the graph of is a subset of the plane.
Suppose is compact, and prove that is continuous on if and only if its graph is compact.
If and if is a function defined on
the restriction of to is the function whose domain of definition is
for Define and on
Prove that is bounded on that is
unbounded in every neighborhood of and that
is not continuous at
nevertheless, the restrictions of both and to every straight line in
Let be a real uniformly continuous function on the bounded set in
Prove that is bounded on
Show that the conclusion is false if boundedness of is omitted from the hypothesis.
Show that the requirement in the definition of uniform continuity can be rephrased as follows, in terms of diameters of sets: To every
there exists a such that
for all with
Complete the details of the following alternative proof of Theorem 4.19: If is not uniformly continuous, then for some
there are sequences
in such that
Use Theorem 2.37 to obtain a contradiction.
Suppose is a uniformly continuous mapping of a metric space into a metric space
and prove that is a Cauchy sequence in
for every Cauchy sequence in
Use this result to give an alternative proof of the theorem stated in Exercise 13.
A uniformly continuous function of a uniformly continuous function is uniformly continuous.
State this more precisely and prove it.
Let be a dense subset of a metric space and let
be a uniformly continuous real function defined on Prove that
has a continuous extension from to (see Exercise 5 for terminology). (Uniqueness follows from Exercise 4.)
Hint: For each and each positive integer let
be the set of all
Use Exercise 9 to show that the intersection of the closures of the sets
consists of a single point, say of
Prove that the function so defined on
is the desired extension of
Could the range space be replaced by
By any compact metric space? By any complete metric space? By any metric space?
Let be the closed unit interval. Suppose
is a continuous mapping of into Prove that
for at least one
Call a mapping of into open if
is an open set in whenever is an open set in
Prove that every continuous open mapping of into is monotonic.
Let denote the largest integer contained in that is,
is the integer such that
denote the fractional part of
What discontinuities do the functions and
Let be a real function defined on
Prove that the set of points at which has a simple discontinuity is at most countable. Hint: Let
be the set on which
With each point of associate a triple
of rational numbers such that
The set of all such triples is countable. Show that each triple is associated with at most one point of
Deal similarly with the other possible types of simple discontinuities.
Every rational can be written in the form
where and and are integers without
any common divisors. When we take
Consider the function defined on by
Prove that is continuous at every irrational point, and that has a simple discontinuity at every rational point.
Suppose is a real function with domain which has the intermediate value property: If
then for some between
Suppose also, for every rational that the set of all with
Prove that is continuous.
Hint: If but
for some and all then
Find a contradiction. (N. J. Fine, Amer. Math. Monthly, vol. 73, 1966, p. 782.)
If is a nonempty subset of a metric space define the distance from
Hint: so that
Prove that if and only if
Prove that is a uniformly continuous function on
by showing that
Suppose and are disjoint sets in a metric space
is compact, is closed. Prove that there exists
such that if
Hint: is a continuous positive function on
Show that the conclusion may fail for two disjoint closed sets if neither is compact.
Let and be disjoint nonempty closed sets in a metric space and define
Show that is a continuous function on whose range lies in
that precisely on and
This establishes a converse of Exercise 3: Every closed set is
for some continuous real on
show that and are open and disjoint, and that
(Thus pairs of disjoint closed sets in a metric space can be covered by pairs of disjoint open sets. This property of metric spaces is called normality.)
A real-valued function defined in is said to be
Prove that every convex function is continuous. Prove that every increasing convex function of a convex function is convex. (For example, if
is convex, so is
If is convex in and if
Assume that is a continuous real function defined in such that
Prove that is convex.
define to be the set of all sums
If is compact and is closed in
prove that is closed.
Hint: Take put
the set of all
Then and are disjoint. Choose as in Exercise 21. Show that the open ball with center
and radius does not intersect
Let be an irrational real number. Let be the set of all integers, let
be the set of all with
and are closed subsets of whose sum
is not closed, by showing that
is a countable dense subset of
Suppose are metric spaces, and is compact. Let
map into let be a continuous one-to-one
mapping of into and put
Prove that is uniformly continuous if is uniformly continuous.
Hint: has compact domain
Prove also that is continuous if is continuous.
Show (by modifying Example 4.21, or by finding a different example) that the compactness of cannot be omitted from the hypotheses, even when
and are compact.