414/514 Simple examples of Baire class one functions

Recall that a real-valued function defined on an interval is (in) Baire class one () iff it is the pointwise limit of continuous functions.

Examples are continuous functions, of course, but functions in do not need to be continuous. An easy example is the function given by if and if . This is the pointwise limit of the functions . By the way, an easy modification of this example shows that any function that is zero except at finitely many points is in .

Step functions are another source of examples. Suppose that and that is constant on each . Then is the pointwise limit of the functions , defined as follows: Fix a decreasing sequence converging to , with and for all . Now define as the restriction of to

,

and let extend by joining consecutive endpoints of the components of its domain with straight segments.

An important source of additional examples is the class of derivatives. Suppose and for all . This is the pointwise limit of the functions given by

This simple construction does not quite work if is defined on a bounded interval (as may fall outside the interval for some values of ). We can modify this easily by using straight segments as in the case of step functions: Say . For large enough so , define as above for , and now set and extend linearly in the interval .

Additional examples can be obtained by observing, first, that is a real vector space, and second, that it is closed under uniform limits (the latter is not quite obvious). This gives us, for instance, that all monotone functions are in , since monotone functions are the uniform limit of step functions on bounded intervals: Given an increasing , let . It follows that all functions of bounded variation are in , since any such function is the difference of two increasing functions.

Another interesting source of examples is characteristic functions. Given , the function is in iff is both an and a set.

On the other hand, is not in , since it is discontinuous everywhere while Baire class one functions are continuous on a comeager set.

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6 Responses to 414/514 Simple examples of Baire class one functions

Sorry I am a bit confused .. Is this saying that we are setting f_n(b) = f(b) = g'(b)? So by taking it as the derivative, it approaches b but never passes it, or touches it?

[…] 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 is an interval, the function is (in) Baire class two () iff it is the pointwise limit of a sequence of Baire one functions. […]

[…] 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 […]

This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]

One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on $X$ such that $|X|

A solution can be obtained as suggested by Keith Conrad in the comments, via Chebotarëv's theorem. Details can be found in $\S3.4$ of Coloring the $n$-Smooth Numbers with $n$ Colors Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach The Electronic Journal of Combinatorics 28 (1) (2021), #P1.34, 79 pp. DOI: https://doi.org/10.37236/8492 Many t […]

Recall that the beth ($\beth$) numbers are defined by transfinite recursion as $\beth_0=0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\alpha

No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]

This is a difficult question in general. Ideally, to show that $f$ is analytic at the origin, you show that in a suitable neighborhood of $0$, the error of the $n$-th Taylor polynomial approaches $0$ as $n\to\infty$. For example, for $f(x)=\sin(x)$, any derivative of $f(x)$ is one of $\sin(x)$, $\cos(x)$, $-\sin(x)$, or $-\cos(x)$, and the error given by the […]

To complement Yann's answer: This is a nice problem, the possible length of Borel hierarchies in different spaces or without assuming the axiom of choice. It has been studied in detail by Arnie Miller. See Arnold W. Miller. On the length of Borel hierarchies, Ann. Math. Logic, 16 (3), (1979), 233–267. MR0548475 (80m:04003), Arnold W. Miller. Long Borel […]

This is a good question, because a priori $\mathsf{PA}$ lacks the flexibility of $\mathsf{ZFC}$ that allows us to deal with consistency problems semantically (by building models) and, anyway, the obvious model of most subtheories of $\mathsf{PA}$ is just the standard model. The way this is done in the context of $\mathsf{ZFC}$ is using the reflection theorem […]

Thanks to Stuart Nygard for suggesting the much easier argument for derivatives being than the messier approach I suggested in lecture.

(Proofs of the closure of under uniform limits, of the continuity fact, and of the claim about characteristic functions, will be provided in lecture.)

Sorry I am a bit confused .. Is this saying that we are setting f_n(b) = f(b) = g'(b)? So by taking it as the derivative, it approaches b but never passes it, or touches it?

Hi Monica.

By construction, each is continuous.

For any , if is large enough then . If , letting , we see that for all sufficiently large, , and this expression converges to .

So the only issue with this definition is whether we also have , but we arrange that this happens trivially, by setting for all .

Putting all this together, we see that pointwise.

oh okay thanks Andres!

[…] 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 is an interval, the function is (in) Baire class two () iff it is the pointwise limit of a sequence of Baire one functions. […]

[…] 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 […]