414/514 Homework 2 – Monotone and Baire one functions

This set is due in three weeks, on Monday, November 3, at the beginning of lecture.

1. Let f:[a,b]\to\mathbb R be increasing. We know that f(x-) and f(x+) exist for all x\in[a,b], and that f has at most countably many points of discontinuity, say t_1,t_2,\dots For each i let I_i,J_i be the intervals (f(t_i-),f(t_i)) and (f(t_i),f(t_i+)). Some of these intervals may be empty, but for each i at least one of them is not. (Here we follow the convention that f(a-)=f(a) and f(b+)=f(b).) Let \mathrm{lh}(I) denote the length of the interval I, and say that an interval (\alpha,\beta) precedes a point t iff \beta\le t.

Verify that \sum_i(\mathrm{lh}(I_i)+\mathrm{lh}(J_i))<+\infty and, more generally, for any x,

s(x):=\sum\{\mathrm{lh}(I_i)\mid I_i precedes f(x)\} +\sum\{\mathrm{lh}(J_i)\mid J_i precedes f(x)\}<+\infty.

Define a function f_0:[a,b]\to\mathbb R by setting f_0(x)=f(x)-s(x). Show that f_0 is increasing and continuous.

Now, for each n>0, define f_n:[a,b]\to\mathbb R so that f_n\upharpoonright[a,t_n)=f_{n-1}\upharpoonright[a,t_n), f_n(t_n)=f_{n-1}(t_n)+\mathrm{lh}(I_n), and f_n(x)=f_{n-1}(x)+\mathrm{lh}(I_n)+\mathrm{lh}(J_n) for all x\in(t_n,b]. Show that each f_n is increasing, and its only discontinuity points are t_1,\dots,t_n.

Prove that f_n\to f 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 f:[a,b]\to\mathbb R is such that for all x, we have that f(x-) and f(x+) exist, then f 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 \epsilon>0 there is a step function s such that |f(x)-s(x)|<\epsilon for all x.

To do this, consider the set A=\{x\in[a,b]\mid there is a step function s on [a,x] such that |f(t)-s(t)|<\epsilon for all t\in[a,x]\}.

Show that A is non-empty. Show that if a\le y\le x and x\in A, then also y\in A. This shows that A is an interval {}[a,\alpha) or {}[a,\alpha], with \alpha\le b. Show that in fact the second possibility occurs, that is, \alpha\in A. For this, the assumption that f(\alpha-) exists is useful. Finally, show that \alpha=b. For this, use now the assumption that f(\alpha+) exists.

3. (This problem is optional.) Find a counterexample to the following statement: If f:[a,b]\to\mathbb R is the pointwise limit of a sequence of functions f_1,f_2,\dots, then there is a dense subset X\subseteq [a,b] where the convergence is in fact uniform. What if f and the functions f_n 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 f:[0,1]\to\mathbb R. Let C be the Cantor set in {}[0,1]. Whenever (a,b) is one of the components of the complement of C, we define f(x)=(2(x-a)/(b-a))-1 for x\in[a,b]. For x not covered by this case, we define f(x)=0. Verify that f is a Darboux continuous function, and that it is discontinuous at every point of C.

Verify that f is not of Baire class one, but that there is a Baire class one function that coincides with f except at (some of) the endpoints of intervals [a,b] as above.

Verify that f is in Baire class two.

3 Responses to 414/514 Homework 2 – Monotone and Baire one functions

  1. I have corrected the definition of the function s in problem 1. Thanks to Jeremy Siegert for noticing the typo in the original version, and for noting that an n-1 should be n.

    Thanks to Stuart Nygard for noticing a further typo in question 2 (some x should have been ts). Fixed now.

  2. Jeremy Siegert says:

    In problem 1 we are supposed to show that each f_{n} is discontinuous on the points t_{1},\dots,t_{n-1}. There is no t_{0} based on how we indexed f‘s points of discontinuity, but it looks as though f_{1} is discontinuous at t_{1}. Should it be that each f_{n} is discontinuous on t_{1},\dots,t_{n}?

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