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 […]

Craig: For a while, there was some research on improving bounds on the number of variables or degree of unsolvable Diophantine equations. Unfortunately, I never got around to cataloging the known results in any systematic way, so all I can offer is some pointers to relevant references, but I am not sure of what the current records are. Perhaps the first pape […]

Yes. Consider, for instance, Conway's base 13 function $c$, or any function that is everywhere discontinuous and has range $\mathbb R$ in every interval. Pick continuous bijections $f_n:\mathbb R\to(-1/n,1/n)$ for $n\in\mathbb N^+$. Pick a strictly decreasing sequence $(x_n)_{n\ge1}$ converging to $0$. Define $f$ by setting $f(x)=0$ if $x=0$ or $\pm x_n […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

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 […]

It is easy to see without choice that if there is a surjection from $A$ onto $B$, then there is an injection from ${\mathcal P}(B)$ into ${\mathcal P}(A)$, and the result follows from Cantor's theorem that $B

Only noticed this question today. Although the selected answer is quite nice and arguably simpler than the argument below, none of the posted answers address what appeared to be the original intent of establishing the inequality using the Arithmetic Mean-Geometric Mean Inequality. For this, simply notice that $$ 1+3+\ldots+(2n-1)=n^2, $$ which can be easily […]

First of all, $f(z)+e^z\ne 0$ by the first inequality. It follows that $e^z/(f(z)+e^z)$ is entire, and bounded above. You should be able to conclude from that.

Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$. Of course there are many possible choices […]

I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is no […]

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 […]