## 414/514 Examples of Baire class two functions

November 3, 2014

Previously, we listed some examples of Baire class one functions. Here we do the same for functions in the next class of Baire. Recall that if $I$ is an interval, the function $f:I\to\mathbb R$ is (in) Baire class two ($\mathcal B_2$) iff it is the pointwise limit of a sequence of Baire one functions.

This post comes from an answer I posted on Math.Stackexchange about a year ago.

Here are three examples:

1. Let $C$ be the Cantor set. For each interval $(a,b)$ contiguous to $C$, define $f$ on ${}[a,b]$ by

$f(x)=\frac{2(x-a)}{b-a}-1$,

so $f$ maps the interval to $[-1,1]$. Otherwise, let $f(x)=0$.

2. Write each $x\in(0,1)$ in binary: $x=0.a_1a_2a_3\dots$, not terminating in a string of $1$s, and define

$\displaystyle f(x)=\limsup_{n\to\infty} \frac{a_1+\dots+a_n}n$.

3. Conway’s base 13 function.

The first two examples come from Bruckner’s book Differentiation of real functions. All three are examples of functions that are not derivatives but have the intermediate value property.

The first one is discontinuous precisely at the points of $C$, and it is “almost” Baire class $1$, in that one can turn it into a Baire class $1$ function by only modifying its values (carefully) at the endpoints of intervals contiguous to $C$. But if one does this, then the function no longer has the intermediate value property.

The second function has the property that the image of any subinterval of $(0,1)$, no matter how small, is all of $(0,1)$. The third function is in the same spirit, but it behaves even more dramatically: The image of every open interval is all of $\mathbb R$.

To verify that the functions are indeed in Baire class at most $2$:

1. For example 1, use that the limit of $x^n$ on ${}[0,1]$ is $0$ for $x<1$, and $1$ at $x=1$, to get for each open interval $(a,b)$ contiguous to $C$ a Baire class $1$ function $f_{[a,b]}$ that is zero everywhere except on ${}[a,b]$, where it coincides with $f$. Now use that the sum of finitely many Baire class $1$ functions is Baire class $1$.
2. For example 2, there are several ways to proceed. Here is one, which I do not think is optimal, but (I believe) is correct: Recall that a limsup is the infimum (over $m$) of a supremum (over all $n>m$), so it is enough to see that each $f_m(x)= \sup_{n>m}g_n$ is Baire class $1$, where

$\displaystyle g_n(x)=\frac{a_1+\dots+a_n}n$.

The point is that each $g_n$ has finitely many discontinuities, all of which are jump discontinuities. Any such function is Baire class $1.$ This would appear to mean that $f_m$ is Baire class $2$, but we are saved by noting that $f_m$ is the uniform limit of the $g_n$, $n>m$. (The point is that each Baire class is closed under uniform limits.)

3. The argument for example 3 is similar. (Note that this function is unbounded.)

To see that the functions are not Baire class $1$: The functions in examples 2 and 3 are discontinuous everywhere, but the set of points of continuity of a Baire class $1$ function is dense. For example 1, use Baire’s extension of this result giving us that, in fact, if $f$ is Baire class $1$, then for any perfect set $P$, the set of points of continuity of $f\upharpoonright P$ is comeager relative to $P$. In example 1 this fails (by design) when $P=C$. (All we need is that, for any closed set $D$, the restriction of a Baire one function to $D$ has at least one continuity point on $D$. Baire also showed that this characterizes Baire one functions.)

Example 2 is also discussed in the van Rooij-Schikhof book (see their Exercise 9.M).

To close, let me include some examples that do not have the intermediate value property. Note first that if $A\subseteq\mathbb R$ and $\chi_A$ is its characteristic (or indicator) function, then $\chi_A$ is continuous iff $A=\emptyset$ or $\mathbb R$. More interestingly, $\chi_A$ is Baire class $1$ iff $A$ is both an $F_\sigma$ and a $G_\delta$ set.

Recall that a set is $F_\sigma$ iff it is the countable union of closed sets, and it is $G_\delta$ iff it is the countable intersection of open sets. The notation $F_\sigma$ is pronounced F-sigma. Here, the F is for fermé, “closed” in French, and the $\sigma$ is for somme, French for “sum”, “union”. Similarly, the notation $G_\delta$ stands for G-delta. Here, the G is for Gebiet, German for “area”, “region”— neighborhood—, and the $\delta$ is for Durchschnitt, German for “intersection”.

Note that, in particular, open sets are both: They are clearly $G_\delta$, and any open interval (and therefore, any countable union of open intervals) is a countable union of closed intervals. It follows that closed sets are also both. In particular, the characteristic function of the Cantor set is Baire class $1$. More generally, a function $f$ is Baire class $1$ iff the preimage $f^{-1}(U)$ of any open set is $F_\sigma$.

For the more general case where $A$ is $F_\sigma$ or $G_\delta$, then $\chi_A$ is Baire class $2$. For any $A$ which is either, but not both, $\chi_A$ is an example of a properly Baire class $2$ function. For instance, this is the case with $A=\mathbb Q$. In fact, $\chi_A$ is Baire class $2$ iff $A$ is both an $F_{\sigma\delta}$ and a $G_{\delta\sigma}$ set ($G_{\delta\sigma}$ sets are countable unions of $G_\delta$ sets, that is, countable unions of countable intersections of open sets, and $F_{\sigma\delta}$ sets are countable intersections of $F_\sigma$ sets, that is, countable intersections of countable unions of closed sets).

More generally, $f$ is Baire class $2$ iff for any open $U$, the set $f^{-1}(U)$ is $G_{\delta\sigma}$. For details, and a significant generalization due to Lebesgue, that characterizes each Baire class and relates it to the hierarchy of Borel sets, see section 24 in Kechris’s book Classical descriptive set theory.

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

October 10, 2014

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.

## 414/514 Advanced Analysis (Analysis I) – Syllabus

August 20, 2014

Instructor: Andrés E. Caicedo
Fall 2014

Time: MWF 10:30-11:45 am.
Place: Mathematics building, Room 139.

Contact Information

• Office: 239-A Mathematics building.
• Phone number: (208)-426-1116. (Not very efficient.)
• Office Hours: W 12:00-1:15 pm. (Or by appointment.)
• Email: caicedo@math.boisestate.edu

Text
We will use three textbooks and complement with papers and handouts for topics not covered there.

• MR1886084 (2003e:00005).
Pugh, Charles Chapman
Real mathematical analysis.
Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2002. xii+437 pp.
ISBN: 0-387-95297-7.
• MR0655599 (83j:26001).
van Rooij, A. C. M.; Schikhof, W. H.
A second course on real functions.
Cambridge University Press, Cambridge-New York, 1982. xiii+200 pp.
ISBN: 0-521-23944-3; 0-521-28361-2.
Gelbaum, Bernard R.; Olmsted, John M. H.
Counterexamples in analysis.
Corrected reprint of the second (1965) edition. Dover Publications, Inc., Mineola, NY, 2003. xxiv+195 pp.
ISBN: 0-486-42875-3.

The book by van Rooij and Schikhof will be our primary reference, supplemented naturally by the Counterexamples book.  The book assumes some knowledge beyond what is covered in our undergraduate course Math 314: Foundations of Analysis, and does not cover the theory in dimension $n>1$; for these topics, we will follow Pugh’s text.

Contents
Math 414/514 covers Analysis on Euclidean spaces (${\mathbb R}^n$) with emphasis on the theory in dimension one. The approach is theoretical, as opposed to the more computational approach of calculus, and a certain degree of mathematical maturity is required. The course is cross-listed, and accordingly the level will be aimed at beginning graduate students.

From the Course Description on the Department’s site:

Introduction to fundamental elements of analysis on Euclidean spaces including the basic differential and integral calculus. Topics include: infinite series, sequences and series of function, uniform convergence, theory of integration, implicit function theorem and applications.

Here is a short list of topics we expect to cover. This list may change based on students’ interest:

1. Set theoretic preliminaries.
• Cantor’s approach to infinite cardinalities. Countable vs. uncountable sets. Sets of size continuum. The Bernstein-Cantor-Schröder theorem.
• The axiom of choice. Zorn’s lemma. Countable and dependent choice.
• Transfinite recursion. The first uncountable ordinal $\omega_1$.
2. Axiomatization and construction of the set of reals.
• The least upper bound property; uniqueness of $\mathbb R$ up to isomorphism.
• Dedekind cuts, and complete orders.
• Metric spaces, and Cauchy completions. Banach contraction mapping theorem.
3. Topology on $\mathbb R$.
• Open and closed sets. Compact sets and Cantor sets. Baire space.
• Borel sets. Analytic sets.
• Notions of smallness.
• Meagerness and the Baire category theorem. The Baire-Cantor stationary principle.
• Sets of Jordan content zero and of measure zero.
• Introduction to the theory of strong measure zero sets.
4. Continuity.
• Sets of discontinuity of functions.
• Monotonicity. Functions of bounded variation.
5. Differentiability.
• The problem of characterizing derivatives. Baire class one functions. The intermediate value property. Sets of continuity of derivatives.
• The mean value theorem. L’Hôpital’s rule.
• The dynamics of Newton’s method.
• The Baire hierarchy of functions.
• Continuous nowhere differentiable functions.
6. Power series.
• Real analytic functions. Taylor series.
• $C^\infty$ functions. Zahorsky’s characterization of the sets of points where a $C^\infty$ function fails to be analytic.
7. Integration.
• Riemann integration. Lebesgue’s characterization of Riemann integrability.
• Weierstraß approximation theorem.
• Lebesgue integration. The fundamental theorem of calculus.
• The Henstock-Kurzweil integral. Denjoy’s approach to reconstructing primitives.
8. Introduction to multivariable calculus.
• (Frechet) derivatives.
• The inverse and implicit function theorems.