This set is due in three weeks, on Monday, November 3, at the beginning of lecture.
1. Let be increasing. We know that and exist for all , and that has at most countably many points of discontinuity, say For each let be the intervals and . Some of these intervals may be empty, but for each at least one of them is not. (Here we follow the convention that and .) Let denote the length of the interval , and say that an interval precedes a point iff .
Verify that and, more generally, for any ,
precedes precedes .
Define a function by setting . Show that is increasing and continuous.
Now, for each , define so that , , and for all . Show that each is increasing, and its only discontinuity points are .
Prove that uniformly.
Use this to provide a (new) proof that increasing functions are in Baire class one.
2. Solve exercise 3.Q in the van Rooij-Schikhof book: If is such that for all , we have that and exist, then is the uniform limit of a sequence of step functions. The approach suggested in the book is the following:
Show that it suffices to argue that for every there is a step function such that for all .
To do this, consider the set there is a step function on such that for all .
Show that is non-empty. Show that if and , then also . This shows that is an interval or , with . Show that in fact the second possibility occurs, that is, . For this, the assumption that exists is useful. Finally, show that . For this, use now the assumption that exists.
3. (This problem is optional.) Find a counterexample to the following statement: If is the pointwise limit of a sequence of functions , then there is a dense subset where the convergence is in fact uniform. What if and the functions are continuous? Can you find a (reasonable) weakening of the statement that is true?
4. (This is example 1.1 in Andrew Bruckner’s Differentiation of real functions, CRM monograph series, AMS, 1994. MR1274044 (94m:26001).) We want to define a function . Let be the Cantor set in . Whenever is one of the components of the complement of , we define for . For not covered by this case, we define . Verify that is a Darboux continuous function, and that it is discontinuous at every point of .
Verify that is not of Baire class one, but that there is a Baire class one function that coincides with except at (some of) the endpoints of intervals as above.
Verify that is in Baire class two.