Neugebauer’s theorem

December 30, 2014

The problem of characterizing when a given function is a derivative has been studied at least since 1911 when Young published a paper recognizing its relevance:

W. H. Young. A note on the property of being a differential coefficient, Proc. Lond. Math. Soc., 9 (2), (1911), 360–368.

The paper opens with the following comment (Young refers to derivatives as “differential coefficients”):

Recent research has provided us with a set of necessary and sufficient conditions that a function may be an indefinite integral, in the generalised sense, of another function, and the way has thus been opened to important developments. The corresponding, much more difficult, problem of determining necessary and sufficient conditions that a function may be a differential coefficient, has barely been mooted; indeed, though we know a number of necessary conditions, no set even of sufficient conditions has to my knowledge ever been formulated, except that involved in the obvious statement that a continuous function is a differential coefficient. 

It is more or less understood nowadays that no completely satisfactory characterization is possible. We know that derivatives are Darboux continuous (that is, they satisfy the intermediate value property), and are Baire one functions (that is, they are the pointwise limit of a sequence of continuous functions). But this is not enough: For instance, any function f such that f(x)=\sin(1/x) for x\ne 0, and f(0)\in[-1,1] is Darboux continuous and Baire one, but only the function with f(0)=0 is a derivative.

Andrew Bruckner has written excellent surveys on derivatives and the characterization problem. See for instance:

Andrew M. Bruckner and John L. Leonard. Derivatives. Amer. Math. Monthly, 73 (4, part II), (1966), 24–56. MR0197632 (33 #5797).

Andrew M. Bruckner. Differentiation of real functions. Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6. MR1274044 (94m:26001).

Andrew M. Bruckner. The problem of characterizing derivatives revisited. Real Anal. Exchange, 21 (1),  (1995/96), 112–133. MR1377522 (97g:26004).

Here I want to discuss briefly a characterization obtained by Neugebauer, see

Christoph Johannes Neugebauer. Darboux functions of Baire class one and derivatives. Proc. Amer. Math. Soc., 13 (6), (1962), 838–843. MR0143850 (26 #1400).

For concreteness, I will restrict discussion to functions f:[0,1]\to\mathbb R, although this makes no real difference. Whenever an interval is mentioned, it is understood to be nondegenerate. For any closed subinterval I\subseteq [0,1], we write I^\circ for its interior, and \mathrm{lh}(I) for its length. Given a point x\in[0,1], we write I\to x to indicate that \mathrm{lh}(I)\to 0 and x\in I.

Theorem (Neugebauer). A function f:[0,1]\to\mathbb R is a derivative iff to each closed subinterval I of {}[0,1] we can associate a point x_I\in I^\circ in such a way that the following hold: 

  1. For all x\in[0,1], if I\to x, then f(x_I)\to f(x), and 

  2. For all  closed subintervals I,I_1,I_2 of {}[0,1], if I=I_1\cup I_2 and {I_1}^\circ\cap {I_2}^\circ=\emptyset, then \mathrm{lh}(I)f(x_I)=\mathrm{lh}(I_1)f(x_{I_1})+\mathrm{lh}(I_2)f(x_{I_2}).

Read the rest of this entry »

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


      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 1s, 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.

314 – Foundations of Analysis – Syllabus

January 20, 2014

Math 314: Foundations of Analysis.

Andrés E. Caicedo.
Contact Information: See here.
Time: TTh 12:00 – 1:15 pm.
Place: Mathematics Building, Room 139.
Office Hours: Th, 1:30 – 3:00 pm, or by appointment. (If you need an appointment, email me a few times/dates that may work for you, and I’ll get back to you).

Textbook: Stephen Abbott. Understanding Analysis. Springer-Verlag, Undergraduate Texts in Mathematics, 2001; 257 pp. ISBN-10: 0387950605. ISBN-13: 978-0387950600.

Here is the publisher’s page. Additional information is available from the author’s page. Review (MR1807438 (2001m:26001)) by Robert Gardner Bartle at MathSciNet. Review by Jeffrey Nunemacher at the American Mathematical Monthly, Vol. 118, No. 2 (February 2011), pp. 186-189.

I will mention additional references, and provide handouts of additional material, as needed.

Contents: The department’s course description reads:

The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.

I strongly suggest you read the material ahead of our meetings, and work on it frequently. You may find some of the topics challenging. If so, here is some excellent advice by Faulkner (from an interview at The Paris Review):

FaulknerPersonally, I find the topics we will study beautiful, and I hope you enjoy learning it as much as I did.

Please bookmark this post. I update it frequently with detailed week-to-week descriptions.

Detailed day to day description and homework assignments. All problems are from Abbott’s book unless otherwise explicitly specified:

  • January 21 – 30. Chapter 1. The real numbers. Irrationality. Completeness. Countable and uncountable sets.
  • January 21. Functions. Mathematical induction and the well-ordering principle.
  • January 23. Sets, logic, quantifiers. Completeness.
  • January 28. Completeness. Countable and uncountable sets. I recommend you read Errol Morris‘s essay on Hypassus of Metapontum, the apparent discoverer of the irrationality of \sqrt2.
  • January 30. Comparing infinities. Counting the rationals. I recommend the following two papers on this topic: 1 and 2. Office hours this week will be on Friday, 11:45-1:15.

Homework set 1 (Due February 4). Exercises 1.2.1, 1.2.2, 1.2.7, 1.2.8, 1.2.10;;, 1.4.11 1.4.13; 1.5.3, 1.5.4, 1.5.9. See below for the required format.

Solution to 1.2.1.

  • February 4 – 20. Chapter 2. Sequences and series. Limits. Cauchy sequences. Infinite series. Riemann‘s rearrangement theorem.
  • February 4. Rearrangements of infinite series, limits of sequences. Homework 1 is due today.
  • February 6. Limit theorems.
  • February 11. Limit theorems continued. Infinite series.
  • February 13. Monotone convergence. The BolzanoWeierstrass theorem.
  • February 18. The Bolzano-Weierstrass theorem continued. Absolute and conditional convergence. Cauchy sequences.
  • February 20. Riemann’s rearrangement’s theorem, and extensions (see here and here). The interesting paper by Marion Scheepers mentioned on the second of those links can be found here.
  • Additional topics: Products of series. Double series.

Homework set 2 (Due February 25). Exercises 2.2.1, 2.2.2, 2.2.5, 2.2.7, 2.3.2, 2.3.3, 2.3.6, 2.3.7, 2.3.9, 2.3.11, 2.4.2, 2.4.4, 2.4.5, 2.5.3, 2.5.4, 2.6.1, 2.6.3, 2.6.5, 2.7.1, 2.7.4, 2.7.6, 2.7.9, 2.7.11. See below for the required format.

  • February 25 – March 6. Chapter 3. Basic topological notions: Open sets. Closed, compact, and perfect sets. The Cantor set. Connectedness. The Baire category theorem.
  • February 25. The Cantor set. Open and closed sets.
  • February 27. Open and closed sets, continued. Extra credit problem: Find a set of reals such that we can obtain 14 different sets by applying to it (any combination of) the operations of complementation and closure. Kuratowski showed that 14 is the largest number that can be obtained that way, you are welcome to also try to show that. (See here.)
  • March 4. Open covers, compact sets. Perfect sets. Connectedness.
  • March 6. The Baire category theorem.
  • Additional topics: The study of closed sets of reals naturally leads to the Cantor-Bendixson derivative, and the Cantor-Baire stationary principle (See here for Ivar Otto Bendixson). A nice reference is Alekos Kechris‘s book, Classical descriptive set theory. For the Baire category theorem and basic applications, I recommend the beginning of John Oxtoby‘s short book, Measure and category. See also the nice paper Subsum Sets: Intervals, Cantor Sets, and Cantorvals by Zbigniew Nitecki, downloadable at the arXiv.

Homework set 3 (Due March 11). Exercises 3.2.1, 3.2.3, 3.2.7, 3.2.9, 3.2.11, 3.2.12, 3.2.14, 3.3.2,, 3.3.9, 3.3.10, 3.4.2, 3.4.4, 3.4.5,,

Solution to 3.3.6.

  • March 11 – March 20. Chapter 4. Limits and continuity: “Continuous” limits. Continuity of functions. The interaction of continuity and compactness.  The intermediate value theorem.
  • March 11. The concept of function. Dirichlet‘s and Thomae‘s examples. Definition of limit and basic properties.
  • March 13. Properties of limits (continued). Definition of continuity and basic properties.
  • March 18. Applications of continuity: The intermediate value property. Banach‘s fixed point theorem.
  • March 20.  Continuity and compactness. Uniform continuity. Sets of discontinuity of functions.
  • Additional topics: The history of the concept of function is very interesting. The intermediate value property also has a curious history. Apparently, for a while it was expected that it sufficed to characterize continuity. Bolzano’s original paper is fairly accessible. A particularly interesting continuous function is the Cantor function, also called the devil’s staircase. The topic of fixed points (Exercise 4.5.7) leads to a beautiful theorem of Sharkovski, on the possible periods of continuous functions (See here for Oleksandr Mykolaiovych Sharkovsky).

Homework set 4 (Due April 1st). Exercises 4.2.1, 4.2.4, 4.2.6, 4.2.7, 4.3.1, 4.3.3, 4.3.4, 4.3.6,, 4.3.12, 4.4.1, 4.4.4, 4.4.6, 4.4.9, 4.4.10, 4.4.13, 4.5.2, 4.5.4, 4.5.7.

  • April 1 – April 10. Chapter 5. Derivatives: What is a derivative? Differentiability and continuity. Darboux theorem. The mean value theorem. Nowhere differentiable functions.
  • April 1. Sets of discontinuity of functions. Definition of derivative, basic properties. Baire class 1 functions.
  • April 3. Darboux theorem (the intermediate value property).
  • April 8. Rolle‘s theorem. The mean value theorem. L’Hôpital’s rule (see here for Guillaume de l’Hôpital).
  • April 10. Continuous nowhere differentiable functions. Weierstrass function. Proper understanding of this topic requires the notion of uniform convergence, that we will discuss in Chapter 6.
  • Supplemental reading: This is a very useful exercise to review the notions of continuity and uniform continuity. For more on the Baire classes of functions, I recommend Kechris’s book on Classical descriptive set theory. The problem of characterizing which functions are derivatives has led to a significant amount of research; these two notes (by Andrew Bruckner, and by Bruckner and J. L. Leonard) discuss some details: 1, 2. On continuous nowhere differentiable functions, the thesis linked to above (by Johan Thim) is a useful resource. Sections 1, 2, 4 of this “quiz” (by Louis A. Talman) complement well the discussion of similar topics in the book. For the history of the mean value theorem, see these slides by Ádám Besenyei.

Homework set 5 (Due April 15). Exercises, 5.3.2, 5.3.3, 5.3.5, 5.3.7.

  • April 15 – April 24. Chapter six: Sequences and series of functions. Pointwise vs. uniform convergence. Uniform convergence, continuity, and differentiability. Power series, Taylor series, C^\infty vs. real analytic.
  • April 15. Pointwise and uniform convergence of sequences of functions. The uniform limit of a sequence of continuous functions is continuous.
  • April 17. Section 6.3: Let (f_n)_{n=1}^\infty be a sequence of differentiable functions defined on a closed interval, that converges pointwise and such that their derivatives converge uniformly. Then the pointwise limit is indeed uniform, the resulting function is differentiable, and its derivative is the limit of the f_n'.
  • April 22. Series of functions. Weierstrass M-test. Power series.
  • April 24. Power series (continued). Taylor series. Real analytic functions.
  • Supplemental reading: On the topic of analytic vs C^\infty functions, see these two essays by Dave L. Renfro: 1, 2. The result of section 6.3 is false if we ask that the sequence of functions f_n converges uniformly while their derivatives converge pointwise. Darji in fact proved that we can have the limit of the f_n be a differentiable function whose derivative disagrees everywhere with the limit of the derivatives. See here. On Formal power series and applications in combinatorics, I recommend the nice paper by Ivan Niven on this topic. For more on real analytic functions, see the first two chapters of the book A primer of real analytic functions, by Steven Krantz and Harold Parks.

Homework set 6 (Due April 29). Exercises 6.2.1, 6.2.5, 6.2.8, 6.2.13, 6.2.15, 6.2.16, 6.3.1, 6.3.4, 6.4.1,, 6.5.1, 6.5.2, 6.6.1, 6.6.6.

  • April 29 – May 8. Chapter seven: The Riemann integral. Darboux’s characterization. Basic properties. The fundamental theorem of calculus. Lebesgue‘s criterion.
  • April 29. Darboux’s approach to the Riemann integral in terms of upper and lower sums. Continuous functions are integrable.
  • May 1. Basic properties of the integral, integrable discontinuous functions. A theorem on uniform convergence ensuring that the integral of a limit is the limit of the integrals.
  • May 6. The fundamental theorem of calculus. Sets of measure zero.
  • May 8. Lebesgue’s characterization of Riemann integrable functions.
  • Supplemental reading: For the interesting history of the early development of the Riemann integral, I suggest the first two chapters of Lebesgue’s theory of integration, by Thomas Hawkins.

Homework set 7 (Due May 13 at 10:30). Exercises 7.2.2, 7.2.5, 7.2.6, 7.3.1, 7.3.3, 7.3.6, 7.4.2, 7.4.4, 7.4.6, 7.5.1, 7.5.4, 7.5.10.

Group project due May 15 at 10:30.

A multiple-choice quiz, by Vicky Neale.


Grading: Based on homework. There will also be a group project, that will count as much as two homework sets. I expect there will be no exams, but if we see the need, you will be informed reasonably in advance.

There is bi-weekly homework, due Tuesdays at the beginning of lecture; you are welcome to turn in your homework early, but I will not accept homework past Tuesdays at 12:05 pm, or grant extensions. The homework covers some routine and some more challenging exercises related to the topics covered in the past two weeks (roughly, one homework set per chapter). It is a good idea to work daily on the homework problems corresponding to the material covered that day.

You are encouraged to work in groups and to ask for help. However, the work you turn in should be written on your own. Give credit as appropriate: Make sure to list all books, websites, and people you collaborated with or consulted while working on the homework. If relevant, indicate what software packages you used, and include any programs you may have written, or additional data.

Your homework must follow the format developed by the mathematics department at Harvey Mudd College. You will find that format at this link. If you do not use this style, unfortunately your homework will be graded as 0. In particular, please make sure that what you turn in is not your scratch work but the final product. Include partial attempts whenever you do not have a full solution.

I may ask you to meet with me to discuss details of sets, and I suggest that before you turn in your work, you make a copy of it, so you can consult it if needed.

I post links to supplementary material on Google+. Circle me and let me know if you are interested, and I’ll add you to my Analysis circle.


Analysis – Some remarks on continued fractions

September 18, 2013

Most introductory books on number theory have at least one section on the theory of continued fractions. I suggest

William Stein. Elementary number theory: primes, congruences, and secrets. A computational approach.  Undergraduate Texts in Mathematics, Springer, New York, 2009. MR2464052 (2009i:11002).

downloadable from his webpage. Chapter 5 is on continued fractions. Of interest to us is the proof that for any integer a_0 and any infinite sequence of positive integers a_1,a_2,\dots the corresponding continued fraction \displaystyle {}[a_0;a_1,a_2,\dots]=a_0+\frac1{a_1+\frac1{a_2+\dots}} converges to an irrational number. This gives us an explicit bijection between the space of irrationals \mathbb R\setminus\mathbb Q and Baire space, \mathbb N^{\mathbb N}. The bijection is actually a homeomorphism, so the two spaces are equal, as topological spaces. This was first noted by Baire, in

René Baire. Sur la représentation des fonctions discontinues. Deuxième partie, Acta Math. 32 (1), (1909), 97–176. MR1555048.

For a very nice alternative proof of the homeomorphism that does not involve continued fractions, see Chapter 1 of

Arnold W. Miller. Descriptive set theory and forcing. How to prove theorems about Borel sets the hard way. Lecture Notes in Logic, 4. Springer-Verlag, Berlin, 1995.MR1439251 (98g:03119).

Arnie’s homeomorphism also has the nice feature of being an explicit order preserving bijection between \mathbb Z^{\mathbb N}, ordered lexicographically, and the set of irrationals.

In William’s book, you may also want to look at section 5.4, where a proof is provided of Euler’s theorem from 1737 that the continued fraction of e is given by


(No such nice pattern is known for \pi.)

A few nice additional results on continued fractions, with references, are given in this blog entry.

A fascinating open problem, due to Zaremba from 1972, is discussed in

Alex Kontorovich. From Apollonius to Zaremba: local-global phenomena in thin orbits. Bull. Amer. Math. Soc. (N.S.), 50 (2), (2013), 187–228. MR3020826.

To state the problem, fix a positive integer n, and consider the set

\displaystyle N_n=\{m\mid \mbox{ there is an }r\mbox{ such that }1\le r\le m, \displaystyle \mathrm{gcd}(m,r)=1,\mbox{ and if }\frac mr=[a_0;a_1,\dots,a_k], \displaystyle \mbox{ then }a_i\le n\mbox{ for all }i\le k\}.

For example, N_1 consists of all numerators appearing in finite continued fractions consisting solely of ones: {}[1],[1;1],[1;1,1],[1;1,1,1],\dots, that is, \displaystyle 1,2,\frac32,\frac53,\frac85,\dots, so N_1 is the set of positive Fibonacci numbers.

Zaremba’s conjecture is that N_5=\mathbb N^+. In fact, much more is expected to be true. For example, Hensley conjectured in 1996 that N_2 contains all positive integers, with finitely many exceptions. The best result to date is that N_{50} contains almost all positive integers, in the sense that

\displaystyle \lim_{n\to\infty}\frac{N_{50}\cap[1,n]}n=1.

For this, see the announcement,

Jean Bourgain, and Alex Kontorovich. On Zaremba’s conjecture, C. R. Math. Acad. Sci. Paris, 349 (9-10), (2011), 493–495. MR2802911 (2012e:11012),

and the preprint

Jean Bourgain, and Alex Kontorovich. On Zaremba’s conjecture. ArXiv:1107.3776.

(Of immediate interest to us is the fact described in Kontorovich’s survey that for n\ge2, the set of infinite continued fractions


where 1\le a_i\le n for all i is a Cantor set.)