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 be increasing. We know that and exist for all , and that has at most countably many points of discontinuity, say For each let be the intervals and . Some of these intervals may be empty, but for each at least one of them is not. (Here we follow the convention that and .) Let denote the length of the interval , and say that an interval precedes a point iff .

Verify that and, more generally, for any ,

precedes precedes .

Define a function by setting . Show that is increasing and continuous.

Now, for each , define so that , , and for all . Show that each is increasing, and its only discontinuity points are .

Prove that 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 is such that for all , we have that and exist, then 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 there is a step function such that for all .

To do this, consider the set there is a step function on such that for all .

Show that is non-empty. Show that if and , then also . This shows that is an interval or , with . Show that in fact the second possibility occurs, that is, . For this, the assumption that exists is useful. Finally, show that . For this, use now the assumption that exists.

3.(This problem is optional.) Find a counterexample to the following statement: If is the pointwise limit of a sequence of functions , then there is a dense subset where the convergence is in fact uniform. What if and the functions 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 . Let be the Cantor set in . Whenever is one of the components of the complement of , we define for . For not covered by this case, we define . Verify that is a Darboux continuous function, and that it is discontinuous at every point of .

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

Verify that is in Baire class two.

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3 Responses to 414/514 Homework 2 – Monotone and Baire one functions

I have corrected the definition of the function in problem 1. Thanks to Jeremy Siegert for noticing the typo in the original version, and for noting that an should be .

Thanks to Stuart Nygard for noticing a further typo in question 2 (some should have been s). Fixed now.

In problem 1 we are supposed to show that each is discontinuous on the points . There is no based on how we indexed ‘s points of discontinuity, but it looks as though is discontinuous at . Should it be that each is discontinuous on ?

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

Ignas: It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here. […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

This is a beautiful and truly fundamental result, and so there are several good quality presentations. Try MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3, or any of the newer editions (the 2003 second ed […]

Given any field automorphism of $\mathbb C$, the rational numbers are fixed. In fact, any number that is explicitly definable in $\mathbb C$ (in the first order language of fields) is fixed. (Actually, this means that we can only ensure that the rationals are fixed, I expand on this below.) Any construction of a wild automorphism uses the axiom of choice. Se […]

The Milner-Rado paradox is only a paradox in the traditional sense of the word: there are no inconsistencies here, but rather the result is (perhaps naively) seen as counter-intuitive. There are two reasons here: First, if $\alpha=\bigcup_{n

The question immediately reminded me of this. Here is an argument following the same basic idea at the beginning of that argument: First, consider $B=\{x\in A\mid A\cap(-\infty,x]$ is countable$\}$, and note that $B$ itself is countable: The point is that if $x\in B$ then $A\cap(-\infty,x]\subseteq B$. Now, if $B\ne\emptyset$, let $t=\sup B$, fix an increasi […]

I have corrected the definition of the function in problem 1. Thanks to Jeremy Siegert for noticing the typo in the original version, and for noting that an should be .

Thanks to Stuart Nygard for noticing a further typo in question 2 (some should have been s). Fixed now.

In problem 1 we are supposed to show that each is discontinuous on the points . There is no based on how we indexed ‘s points of discontinuity, but it looks as though is discontinuous at . Should it be that each is discontinuous on ?

Yes, exactly.