Baby Rudin Problems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 19 March 2014

Chapter 4

1. Suppose $f$ is a real function defined on $R^1$ which satisfies $$\underset{h\to 0}{\text{lim}}[f(x+h)-f(x-h)]=0$$ 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 $$f\left(\bar{E}\right)\subset \overline{f\left(E\right)}$$ for every set $E\subset X\text{.}$ $\text{(}\bar{E}$ denotes the closure of $E\text{.)}$ Show, by an example, that $f\left(\bar{E}\right)$ can be a proper subset of $\overline{f\left(E\right)}\text{.}$
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 $(x,f\left(x\right)),$ 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: $f(0,0)=g(0,0)=0,$ $f(x,y)=x{y}^2/(x^2+y^4),$ $g(x,y)=x{y}^2/(x^2+y^6)$ if $(x,y)\ne (0,0)\text{.}$ Prove that $f$ is bounded on $R^2,$ that $g$ is unbounded in every neighborhood of $(0,0),$ and that $f$ is not continuous at $(0,0)\text{;}$ 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 $\text{diam}\hspace{0.17em}f\left(E\right)<\epsilon$ for all $E\subset X$ with $\text{diam}\hspace{0.17em}E<\delta \text{.}$
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 $d_X(p_n,q_n)\to 0$ but $d_Y(f\left(p_n\right),f\left(q_n\right))>\epsilon \text{.}$ 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 $d(p,q)<1/n\text{.}$ 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 $I=[0,1]$ 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 $(a,b)\text{.}$ 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 $f(x-)<f(x+)\text{.}$ With each point $x$ of $E,$ associate a triple $(p,q,r)$ of rational numbers such that
    1. $f(x-)<p<f(x+),$
    2. $a<q<t<x$ implies $f\left(t\right)<p,$
    3. $x<t<r<b$ 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 $$f\left(x\right)=\{\begin{array}{cc}0& \left(x\hspace{0.17em}\text{irrational}\right),\\ \frac{1}{n}& (x=\frac{m}{n})\text{.}\end{array}$$ 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 $f\left(a\right)<c<f\left(b\right),$ 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 $${\rho }_E\left(x\right)=\underset{x\in E}{\text{inf}}\hspace{0.17em}d(x,z)\text{.}$$
    1. Prove that ${\rho }_E\left(x\right)=0$ if and only if $x\in \bar{E}\text{.}$
    2. Prove that ${\rho }_E$ is a uniformly continuous function on $X,$ by showing that $$|{\rho }_E\left(x\right)-{\rho }_E\left(y\right)|\le d(x,y)$$ for all $x\in X,$ $y\in X\text{.}$
    Hint: ${\rho }_E\left(x\right)\le d(x,z)\le d(x,y)+d(y,z),$ so that $${\rho }_E\left(x\right)\le d(x,y)+{\rho }_E\left(y\right)\text{.}$$
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 $d(p,q)>\delta$ 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 $$f\left(p\right)=\frac{{\rho }_A\left(p\right)}{{\rho }_A\left(p\right)+{\rho }_B\left(p\right)}(p\in X)\text{.}$$ Show that $f$ is a continuous function on $X$ whose range lies in $[0,1],$ 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 $$V=f^{-1}\left([0,\frac{1}{2}]\right),W=f^{-1}\left((\frac{1}{2},1]\right),$$ 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 $(a,b)$ is said to be convex if $$f(\lambda x+(1-\lambda )y)\le \lambda f\left(x\right)+(1-\lambda )f\left(y\right)$$ whenever $a<x<b,$ $a<y<b,$ $0<\lambda <1\text{.}$ 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 $(a,b)$ and if $a<s<t<u<b,$ show that $$\frac{f\left(t\right)-f\left(s\right)}{t-s}\le \frac{f\left(u\right)-f\left(s\right)}{u-s}\le \frac{f\left(u\right)-f\left(t\right)}{u-t}\text{.}$$
24. Assume that $f$ is a continuous real function defined in $(a,b)$ such that $$f\left(\frac{x+y}{2}\right)\le \frac{f\left(x\right)+f\left(y\right)}{2}$$ for all $x,y\in (a,b)\text{.}$ 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.

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