## 414/514 Homework 2 – Monotone and Baire one functions

This set is due in three weeks, on Monday, November 3, at the beginning of lecture.

1. Let $f:[a,b]\to\mathbb R$ be increasing. We know that $f(x-)$ and $f(x+)$ exist for all $x\in[a,b]$, and that $f$ has at most countably many points of discontinuity, say $t_1,t_2,\dots$ For each $i$ let $I_i,J_i$ be the intervals $(f(t_i-),f(t_i))$ and $(f(t_i),f(t_i+))$. Some of these intervals may be empty, but for each $i$ at least one of them is not. (Here we follow the convention that $f(a-)=f(a)$ and $f(b+)=f(b)$.) Let $\mathrm{lh}(I)$ denote the length of the interval $I$, and say that an interval $(\alpha,\beta)$ precedes a point $t$ iff $\beta\le t$.

Verify that $\sum_i(\mathrm{lh}(I_i)+\mathrm{lh}(J_i))<+\infty$ and, more generally, for any $x$,

$s(x):=\sum\{\mathrm{lh}(I_i)\mid I_i$ precedes $f(x)\}$ $+\sum\{\mathrm{lh}(J_i)\mid J_i$ precedes $f(x)\}<+\infty$.

Define a function $f_0:[a,b]\to\mathbb R$ by setting $f_0(x)=f(x)-s(x)$. Show that $f_0$ is increasing and continuous.

Now, for each $n>0$, define $f_n:[a,b]\to\mathbb R$ so that $f_n\upharpoonright[a,t_n)=f_{n-1}\upharpoonright[a,t_n)$, $f_n(t_n)=f_{n-1}(t_n)+\mathrm{lh}(I_n)$, and $f_n(x)=f_{n-1}(x)+\mathrm{lh}(I_n)+\mathrm{lh}(J_n)$ for all $x\in(t_n,b]$. Show that each $f_n$ is increasing, and its only discontinuity points are $t_1,\dots,t_n$.

Prove that $f_n\to f$ uniformly.

Use this to provide a (new) proof that increasing functions are in Baire class one.

2. Solve exercise 3.Q in the van Rooij-Schikhof book: If $f:[a,b]\to\mathbb R$ is such that for all $x$, we have that $f(x-)$ and $f(x+)$ exist, then $f$ is the uniform limit of a sequence of step functions. The approach suggested in the book is the following:

Show that it suffices to argue that for every $\epsilon>0$ there is a step function $s$ such that $|f(x)-s(x)|<\epsilon$ for all $x$.

To do this, consider the set $A=\{x\in[a,b]\mid$ there is a step function $s$ on $[a,x]$ such that $|f(t)-s(t)|<\epsilon$ for all $t\in[a,x]\}$.

Show that $A$ is non-empty. Show that if $a\le y\le x$ and $x\in A$, then also $y\in A$. This shows that $A$ is an interval ${}[a,\alpha)$ or ${}[a,\alpha]$, with $\alpha\le b$. Show that in fact the second possibility occurs, that is, $\alpha\in A$. For this, the assumption that $f(\alpha-)$ exists is useful. Finally, show that $\alpha=b$. For this, use now the assumption that $f(\alpha+)$ exists.

3. (This problem is optional.) Find a counterexample to the following statement: If $f:[a,b]\to\mathbb R$ is the pointwise limit of a sequence of functions $f_1,f_2,\dots$, then there is a dense subset $X\subseteq [a,b]$ where the convergence is in fact uniform. What if $f$ and the functions $f_n$ are continuous?  Can you find a (reasonable) weakening of the statement that is true?

4. (This is example 1.1 in Andrew Bruckner’s Differentiation of real functions, CRM monograph series, AMS, 1994. MR1274044 (94m:26001).) We want to define a function $f:[0,1]\to\mathbb R$. Let $C$ be the Cantor set in ${}[0,1]$. Whenever $(a,b)$ is one of the components of the complement of $C$, we define $f(x)=(2(x-a)/(b-a))-1$ for $x\in[a,b]$. For $x$ not covered by this case, we define $f(x)=0$. Verify that $f$ is a Darboux continuous function, and that it is discontinuous at every point of $C$.

Verify that $f$ is not of Baire class one, but that there is a Baire class one function that coincides with $f$ except at (some of) the endpoints of intervals $[a,b]$ as above.

Verify that $f$ is in Baire class two.

1. I have corrected the definition of the function $s$ in problem 1. Thanks to Jeremy Siegert for noticing the typo in the original version, and for noting that an $n-1$ should be $n$.
Thanks to Stuart Nygard for noticing a further typo in question 2 (some $x$ should have been $t$s). Fixed now.
In problem 1 we are supposed to show that each $f_{n}$ is discontinuous on the points $t_{1},\dots,t_{n-1}$. There is no $t_{0}$ based on how we indexed $f$‘s points of discontinuity, but it looks as though $f_{1}$ is discontinuous at $t_{1}$. Should it be that each $f_{n}$ is discontinuous on $t_{1},\dots,t_{n}$?