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.
Here are three examples:
- Let be the Cantor set. For each interval contiguous to , define on by
so maps the interval to . Otherwise, let .
- Write each in binary: , not terminating in a string of s, and define
- Conway’s base 13 function.
The first two examples come from Bruckner’s book Differentiation of real functions. All three are examples of functions that are not derivatives but have the intermediate value property.
The first one is discontinuous precisely at the points of , and it is “almost” Baire class , in that one can turn it into a Baire class function by only modifying its values (carefully) at the endpoints of intervals contiguous to . But if one does this, then the function no longer has the intermediate value property.
The second function has the property that the image of any subinterval of , no matter how small, is all of . The third function is in the same spirit, but it behaves even more dramatically: The image of every open interval is all of .
To verify that the functions are indeed in Baire class at most :
- For example 1, use that the limit of on is for , and at , to get for each open interval contiguous to a Baire class function that is zero everywhere except on , where it coincides with . Now use that the sum of finitely many Baire class functions is Baire class .
- For example 2, there are several ways to proceed. Here is one, which I do not think is optimal, but (I believe) is correct: Recall that a limsup is the infimum (over ) of a supremum (over all ), so it is enough to see that each is Baire class , where
The point is that each has finitely many discontinuities, all of which are jump discontinuities. Any such function is Baire class This would appear to mean that is Baire class , but we are saved by noting that is the uniform limit of the , . (The point is that each Baire class is closed under uniform limits.)
- The argument for example 3 is similar. (Note that this function is unbounded.)
To see that the functions are not Baire class : The functions in examples 2 and 3 are discontinuous everywhere, but the set of points of continuity of a Baire class function is dense. For example 1, use Baire’s extension of this result giving us that, in fact, if is Baire class , then for any perfect set , the set of points of continuity of is comeager relative to . In example 1 this fails (by design) when . (All we need is that, for any closed set , the restriction of a Baire one function to has at least one continuity point on . Baire also showed that this characterizes Baire one functions.)
Example 2 is also discussed in the van Rooij-Schikhof book (see their Exercise 9.M).
To close, let me include some examples that do not have the intermediate value property. Note first that if and is its characteristic (or indicator) function, then is continuous iff or . More interestingly, is Baire class iff is both an and a set.
Recall that a set is iff it is the countable union of closed sets, and it is iff it is the countable intersection of open sets. The notation is pronounced F-sigma. Here, the F is for fermé, “closed” in French, and the is for somme, French for “sum”, “union”. Similarly, the notation stands for G-delta. Here, the G is for Gebiet, German for “area”, “region”— neighborhood—, and the is for Durchschnitt, German for “intersection”.
Note that, in particular, open sets are both: They are clearly , and any open interval (and therefore, any countable union of open intervals) is a countable union of closed intervals. It follows that closed sets are also both. In particular, the characteristic function of the Cantor set is Baire class . More generally, a function is Baire class iff the preimage of any open set is .
For the more general case where is or , then is Baire class . For any which is either, but not both, is an example of a properly Baire class function. For instance, this is the case with . In fact, is Baire class iff is both an and a set ( sets are countable unions of sets, that is, countable unions of countable intersections of open sets, and sets are countable intersections of sets, that is, countable intersections of countable unions of closed sets).
More generally, is Baire class iff for any open , the set is . For details, and a significant generalization due to Lebesgue, that characterizes each Baire class and relates it to the hierarchy of Borel sets, see section 24 in Kechris’s book Classical descriptive set theory.