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 […]

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All proofs of the Bernstein-Cantor-Schroeder theorem that I know either directly or with very little work produce an explicit bijection from any given pair of injections. There is an obvious injection from $[0,1]$ to $C[0,1]$ mapping each $t$ to the function constantly equal to $t$, so the question reduces to finding an explicit injection from $C[0,1]$ to $[ […]

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

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Fix a model $M$ of a theory for which it makes sense to talk about $\omega$ ($M$ does not need to be a model of set theory, it could even be simply an ordered set with a minimum in which every element has an immediate successor and every element other than the minimum has an immediate predecessor; in this case we could identify $\omega^M$ with $M$ itself). W […]

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 […]