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 […]

The technique of almost disjoint forcing was introduced in MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104, North-Holland, Amsterdam, 1970. Fix an almost disjoint family $X=(x_\alpha:\alpha

At the moment most of those decisions come from me, at least for computer science papers (those with a 68 class as primary). The practice of having proceedings and final versions of papers is not exclusive to computer science, but this is where it is most common. I've found more often than not that the journal version is significantly different from the […]

The answer is no in general. For instance, by what is essentially an argument of Sierpiński, if $(X,\Sigma,\nu)$ is a $\sigma$-finite continuous measure space, then no non-null subset of $X$ admits a $\nu\times\nu$-measurable well-ordering. The proof is almost verbatim the one here. It is consistent (assuming large cardinals) that there is an extension of Le […]

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Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is negative. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${\mathfrak c}$ (This doesn't matter, all we need is that it is strictly larger. T […]

Yes, the suggested rearrangement converges to 0. This is a particular case of a result of Martin Ohm: For $p$ and $q$ positive integers rearrange the sequence $$\left(\frac{(−1)^{n-1}} n\right)_{n\ge 1} $$ by taking the ﬁrst $p$ positive terms, then the ﬁrst $q$ negative terms, then the next $p$ positive terms, then the next $q$ negative terms, and so on. Th […]

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A simple example is the permutation $\pi$ given by $\pi(n)=n+2$ if $n$ is even, $\pi(1)=0$, and otherwise $\pi(n)=n−2$. It should be clear that $\pi$ is computable and has the desired property. By the way, regarding the footnote: if a bijection is computable, so is its inverse, so $\pi^{-1}$ is computable as well. In general, given a computable bijection $\s […]

The question is asking to find all polynomials $f$ for which you can find $a,b\in\mathbb R$ with $a\ne b$ such that the displayed identity holds. The concrete numbers $a,b$ may very well depend on $f$. A priori, it may be that for some $f$ there is only one pair for which the identity holds, it may be that for some $f$ there are many such pairs, and it may a […]

The reflection principle is a theorem schema in ZFC, meaning that for each formula $\phi(\vec x)$ we can prove in ZFC a version of the principle for $\phi$. In particular, it gives us that if $\phi$ holds (in the universe of sets) then there is some ordinal $\alpha$ such that $V_\alpha\models \phi$. It follows from this that (assuming its consistency) $\math […]

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 […]