## Baby Rudin Problems

Last update: 19 March 2014

## Chapter 4

1. Suppose $f$ is a real function defined on ${R}^{1}$ which satisfies $limh→0 [ f(x+h)- f(x-h) ] =0$ for every $x\in {R}^{1}\text{.}$ Does this imply that $f$ is continuous?
2. If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y,$ prove that $f(E‾)⊂ f(E)‾$ for every set $E\subset X\text{.}$ $\text{(}\stackrel{‾}{E}$ denotes the closure of $E\text{.)}$ Show, by an example, that $f\left(\stackrel{‾}{E}\right)$ can be a proper subset of $\stackrel{‾}{f\left(E\right)}\text{.}$
3. Let $f$ be a continuous real function on a metric space $X\text{.}$ Let $Z\left(f\right)$ (the zero set of $f\text{)}$ be the set of all $p\in X$ at which $f\left(p\right)=0\text{.}$ Prove that $Z\left(f\right)$ is closed.
4. Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y,$ and let $E$ be a dense subset of $X\text{.}$ Prove that $f\left(E\right)$ is dense in $f\left(X\right)\text{.}$ If $g\left(p\right)=f\left(p\right)$ for all $p\in E,$ prove that $g\left(p\right)=f\left(p\right)$ for all $p\in X\text{.}$ (In other words, a continuous mapping is determined by its values on a dense subset of its domain.)
5. If $f$ is a real continuous function defined on a closed set $E\subset {R}^{1},$ prove that there exist continuous real functions $g$ on ${R}^{1}$ such that $g\left(x\right)=f\left(x\right)$ for all $x\in E\text{.}$ (Such functions $g$ are called continuous extensions of $f$ from $E$ to ${R}^{1}\text{.)}$ Show that the result becomes false if the word "closed" is omitted. Extend the result to vector-valued functions. Hint: Let the graph of $g$ be a straight line on each of the segments which constitute the complement of $E$ (compare Exercise 29, Chap. 2). The result remains true if ${R}^{1}$ is replaced by any metric space, but the proof is not so simple.
6. If $f$ is defined on $E,$ the graph of $f$ is the set of points $\left(x,f\left(x\right)\right),$ for $x\in E\text{.}$ In particular, if $E$ is a set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous on $E$ if and only if its graph is compact.
7. If $E\subset X$ and if $f$ is a function defined on $X,$ the restriction of $f$ to $E$ is the function $g$ whose domain of definition is $E,$ such that $g\left(p\right)=f\left(p\right)$ for $p\in E\text{.}$ Define $f$ and $g$ on ${R}^{2}$ by: $f\left(0,0\right)=g\left(0,0\right)=0,$ $f\left(x,y\right)=x{y}^{2}/\left({x}^{2}+{y}^{4}\right),$ $g\left(x,y\right)=x{y}^{2}/\left({x}^{2}+{y}^{6}\right)$ if $\left(x,y\right)\ne \left(0,0\right)\text{.}$ Prove that $f$ is bounded on ${R}^{2},$ that $g$ is unbounded in every neighborhood of $\left(0,0\right),$ and that $f$ is not continuous at $\left(0,0\right)\text{;}$ nevertheless, the restrictions of both $f$ and $g$ to every straight line in ${R}^{2}$ are continuous!
8. Let $f$ be a real uniformly continuous function on the bounded set $E$ in ${R}^{1}\text{.}$ Prove that $f$ is bounded on $E\text{.}$ Show that the conclusion is false if boundedness of $E$ is omitted from the hypothesis.
9. Show that the requirement in the definition of uniform continuity can be rephrased as follows, in terms of diameters of sets: To every $\epsilon >0$ there exists a $\delta >0$ such that $\text{diam} f\left(E\right)<\epsilon$ for all $E\subset X$ with $\text{diam} E<\delta \text{.}$
10. Complete the details of the following alternative proof of Theorem 4.19: If $f$ is not uniformly continuous, then for some $\epsilon >0$ there are sequences $\left\{{p}_{n}\right\},\left\{{q}_{n}\right\}$ in $X$ such that ${d}_{X}\left({p}_{n},{q}_{n}\right)\to 0$ but ${d}_{Y}\left(f\left({p}_{n}\right),f\left({q}_{n}\right)\right)>\epsilon \text{.}$ Use Theorem 2.37 to obtain a contradiction.
11. Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$ and prove that $\left\{f\left({x}_{n}\right)\right\}$ is a Cauchy sequence in $Y$ for every Cauchy sequence $\left\{{x}_{n}\right\}$ in $X\text{.}$ Use this result to give an alternative proof of the theorem stated in Exercise 13.
12. A uniformly continuous function of a uniformly continuous function is uniformly continuous. State this more precisely and prove it.
13. Let $E$ be a dense subset of a metric space $X,$ and let $f$ be a uniformly continuous real function defined on $E\text{.}$ Prove that $f$ has a continuous extension from $E$ to $X$ (see Exercise 5 for terminology). (Uniqueness follows from Exercise 4.) Hint: For each $p\in X$ and each positive integer $n,$ let ${V}_{n}\left(p\right)$ be the set of all $q\in E$ with $d\left(p,q\right)<1/n\text{.}$ Use Exercise 9 to show that the intersection of the closures of the sets $f\left({V}_{1}\left(p\right)\right),f\left({V}_{2}\left(p\right)\right),\dots ,$ consists of a single point, say $g\left(p\right),$ of ${R}^{1}\text{.}$ Prove that the function $g$ so defined on $X$ is the desired extension of $f\text{.}$ Could the range space ${R}^{1}$ be replaced by ${R}^{k}\text{?}$ By any compact metric space? By any complete metric space? By any metric space?
14. Let $I=\left[0,1\right]$ be the closed unit interval. Suppose $f$ is a continuous mapping of $I$ into $I\text{.}$ Prove that $f\left(x\right)=x$ for at least one $x\in I\text{.}$
15. Call a mapping of $X$ into $Y$ open if $f\left(V\right)$ is an open set in $Y$ whenever $V$ is an open set in $X\text{.}$ Prove that every continuous open mapping of ${R}^{1}$ into ${R}^{1}$ is monotonic.
16. Let $\left[x\right]$ denote the largest integer contained in $x,$ that is, $\left[x\right]$ is the integer such that $x-1<\left[x\right]\le x\text{;}$ and let $\left(x\right)=x-\left[x\right]$ denote the fractional part of $x\text{.}$ What discontinuities do the functions $\left[x\right]$ and $\left(x\right)$ have?
17. Let $f$ be a real function defined on $\left(a,b\right)\text{.}$ Prove that the set of points at which $f$ has a simple discontinuity is at most countable. Hint: Let $E$ be the set on which $f\left(x-\right) With each point $x$ of $E,$ associate a triple $\left(p,q,r\right)$ of rational numbers such that
1. $f\left(x-\right)
2. $a implies $f\left(t\right)
3. $x implies $f\left(t\right)>p\text{.}$
The set of all such triples is countable. Show that each triple is associated with at most one point of $E\text{.}$ Deal similarly with the other possible types of simple discontinuities.
18. Every rational $x$ can be written in the form $x=m/n,$ where $n>0,$ and $m$ and $n$ are integers without any common divisors. When $x=0,$ we take $n=1\text{.}$ Consider the function $f$ defined on ${R}^{1}$ by $f(x)= { 0 (x irrational), 1n (x=mn).$ Prove that $f$ is continuous at every irrational point, and that $f$ has a simple discontinuity at every rational point.
19. Suppose $f$ is a real function with domain ${R}^{1}$ which has the intermediate value property: If $f\left(a\right) then $f\left(x\right)=c$ for some $x$ between $a$ and $b\text{.}$ Suppose also, for every rational $r,$ that the set of all $x$ with $f\left(x\right)=r$ is closed. Prove that $f$ is continuous. Hint: If ${x}_{n}\to {x}_{0}$ but $f\left({x}_{n}\right)>r>f\left({x}_{0}\right)$ for some $r$ and all $n,$ then $f\left({t}_{n}\right)=r$ for some ${t}_{n}$ between ${x}_{0}$ and ${x}_{n}\text{;}$ thus ${t}_{n}\to {x}_{0}\text{.}$ Find a contradiction. (N. J. Fine, Amer. Math. Monthly, vol. 73, 1966, p. 782.)
20. If $E$ is a nonempty subset of a metric space $X,$ define the distance from $x\in X$ to $E$ by $ρE(x)= infx∈E d(x,z).$
1. Prove that ${\rho }_{E}\left(x\right)=0$ if and only if $x\in \stackrel{‾}{E}\text{.}$
2. Prove that ${\rho }_{E}$ is a uniformly continuous function on $X,$ by showing that $|ρE(x)-ρE(y)| ≤d(x,y)$ for all $x\in X,$ $y\in X\text{.}$
Hint: ${\rho }_{E}\left(x\right)\le d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right),$ so that $ρE(x)≤d(x,y) +ρE(y).$
21. Suppose $K$ and $F$ are disjoint sets in a metric space $X,$ $K$ is compact, $F$ is closed. Prove that there exists $\delta >0$ such that $d\left(p,q\right)>\delta$ if $p\in K,$ $q\in F\text{.}$ Hint: ${\rho }_{F}$ is a continuous positive function on $K\text{.}$ Show that the conclusion may fail for two disjoint closed sets if neither is compact.
22. Let $A$ and $B$ be disjoint nonempty closed sets in a metric space $X,$ and define $f(p)= ρA(p)ρA(p)+ρB(p) (p∈X).$ Show that $f$ is a continuous function on $X$ whose range lies in $\left[0,1\right],$ that $f\left(p\right)=0$ precisely on $A$ and $f\left(p\right)=1$ precisely on $B\text{.}$ This establishes a converse of Exercise 3: Every closed set $A\subset X$ is $Z\left(f\right)$ for some continuous real $f$ on $X\text{.}$ Setting $V=f-1 ([0,12]), W=f-1 ((12,1]),$ show that $V$ and $W$ are open and disjoint, and that $A\subset V,$ $B\subset W\text{.}$ (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.)
23. A real-valued function $f$ defined in $\left(a,b\right)$ is said to be convex if $f(λx+(1-λ)y) ≤λf(x)+(1-λ) f(y)$ whenever $a $a $0<\lambda <1\text{.}$ Prove that every convex function is continuous. Prove that every increasing convex function of a convex function is convex. (For example, if $f$ is convex, so is ${e}^{f}\text{.)}$ If $f$ is convex in $\left(a,b\right)$ and if $a show that $f(t)-f(s)t-s≤ f(u)-f(s)u-s≤ f(u)-f(t)u-t.$
24. Assume that $f$ is a continuous real function defined in $\left(a,b\right)$ such that $f(x+y2)≤ f(x)+f(y)2$ for all $x,y\in \left(a,b\right)\text{.}$ Prove that $f$ is convex.
25. If $A\subset {R}^{k}$ and $B\subset {R}^{k},$ define $A+B$ to be the set of all sums $\text{x}+\text{y}$ with $\text{x}\in A,$ $\text{y}\in B\text{.}$
1. If $K$ is compact and $C$ is closed in ${R}^{k},$ prove that $K+C$ is closed. Hint: Take $\text{z}\in K+C,$ put $F=\text{z}-C,$ the set of all $\text{z}-\text{y}$ with $\text{y}\in C\text{.}$ Then $K$ and $F$ are disjoint. Choose $\delta$ as in Exercise 21. Show that the open ball with center $\text{z}$ and radius $\delta$ does not intersect $K+C\text{.}$
2. Let $\alpha$ be an irrational real number. Let ${C}_{1}$ be the set of all integers, let ${C}_{2}$ be the set of all $n\alpha$ with $n\in {C}_{1}\text{.}$ Show that ${C}_{1}$ and ${C}_{2}$ are closed subsets of ${R}^{1}$ whose sum ${C}_{1}+{C}_{2}$ is not closed, by showing that ${C}_{1}+{C}_{2}$ is a countable dense subset of ${R}^{1}\text{.}$
26. Suppose $X,Y,Z$ are metric spaces, and $Y$ is compact. Let $f$ map $X$ into $Y,$ let $g$ be a continuous one-to-one mapping of $Y$ into $Z,$ and put $h\left(x\right)=g\left(f\left(x\right)\right)$ for $x\in X\text{.}$ Prove that $f$ is uniformly continuous if $h$ is uniformly continuous. Hint: ${g}^{-1}$ has compact domain $g\left(Y\right),$ and $f\left(x\right)={g}^{-1}\left(h\left(x\right)\right)\text{.}$ Prove also that $f$ is continuous if $h$ is continuous. Show (by modifying Example 4.21, or by finding a different example) that the compactness of $Y$ cannot be omitted from the hypotheses, even when $X$ and $Z$ are compact.