Baire One Functions - Study Guide

BAIRE ONE FUNCTIONS Abstract. This paper gives a general overview of Baire one functions, including examples as well as several interesting properties involving bounds, uniform convergence, continuity, and F sets. We conclude with a result on a characterization of Baire one functions in terms of the notion of first return recoverability, which is a topic of current research in analysis [6]. 1. History Baire one functions are named in honor of Ren´e-Louis Baire (1874-1932), a French mathematician who had research interests in continuity of functions and the idea of limits [1]. The problem of classifying the class of functions that are convergent sequences of continuous functions was first explored in 1897. According to F.A. Medvedev in his book Scenes from the History of Real Functions, Baire was interested in the relationship between the continuity of a function of two variables in each argument separately and its joint continuity in the two variables [2]. Baire showed that every function of two variables that is continuous in each argument separately is not pointwise continuous with respect to the two variables jointly [2]. For instance, consider the function xy . + y2 Observe that for all points (x, y) 6= 0, f is continuous and so, of course, f is separately continuous. Next, we see that, f (x, y) = x2 xy xy = lim =0 2 2 (0,y)→(0,0) x + y 2 +y as (x, y) approaches (0, 0) along the x-axis and y-axis respectively. However, as (x, y) approaches (0, 0) along the line y = x, we have xy 1 lim = . 2 2 (x,x)→(0,0) x + y 2 Hence, f is continuous along the lines y = 0 and x = 0, but is discontinuous overall at (0, 0). It can be shown that f is a Baire one function. It was this idea that motivated the development of the classification lim (x,0)→(0,0) x2 1 2 of Baire class functions, which was later generalized in Baire’s 1899 dissertation Sur les fonctions des variables réelles [2]. In general, a function that is Baire class n, is the pointwise limit of a sequence of Baire class n − 1 functions, where Baire class 0 consists of the set of all continuous functions [3]. For instance, a function that is Baire class one (which we will define in the next section) is the pointwise limit of continuous functions. Functions that are Baire class two are the pointwise limit of a sequence of Baire class one functions, and so on. 2. Preliminaries and Definition of Baire One Before we define what it means for a function to be Baire one, it is necessary to first introduce some background that will be helpful. We assume that the reader has some familiarity with sequences of real numbers and what it means for a sequence to converge. It is possible to make analogous statements in terms of functions. Let {fn } define a sequence of functions whose domain and range is the set of real numbers. We are interested to see if {fn } converges to a specific function at each point in its interval for which it is defined. This type of convergence is known as pointwise convergence and we state it below. Definition 1 (Pointwise Convergence). Let {fn } be a sequence of functions defined on a set D and also let f be a function defined on D. If the sequence {fn (x)} converges to f (x) for each x ∈ D, then {fn } converges pointwise to f on D. In other words, f (x) = limn→∞ fn (x) for all x ∈ I. An example to illustrate the definition is as follows. Let gn : [0, 1] → R be defined as gn (x) = xn . We see that the sequence converges pointwise to  0, for x ∈ [0, 1); g(x) = 1, if x = 1. From Figure 1, we notice that each gn is continuous on the closed interval [0, 1]. Given our definition of pointwise convergence, we now state the formal definition of a Baire one function. Definition 2. [5] Let D ⊆ R. A function f : D → R is called a Baire one function if f is the pointwise limit of a sequence of continuous functions, that is, if there is a sequence {fn } of functions continuous on D such that for every x ∈ D, f (x) = limn→∞ fn (x). The function g above is an example of a Baire one function since each function gn is continuous and converges pointwise to g on [0, 1] as n → ∞. BAIRE ONE FUNCTIONS 3 1.0 0.75 0.5 0.25 0.0 −2 −1 0 1 2 x Figure 1. The function gn , n = 1, 2, 3, 4, 5, 6 3. Examples and Some Properties of Baire One Functions We now present some examples as well as some interesting properties of Baire one functions. As shown in the previous section, a Baire one function does not have to be continuous, however all continuous functions are Baire one by definition. To show this, suppose j is a continuous function and let jn : R → R where jn (x) = j(x) + 1/n, for each n ∈ Z+ . It follows that j(x) = limn→∞ jn (x). Another question that we might ask is whether or not we can we extend the domain of a Baire one function to the entire real line. The answer is yes. Let kn : R → R be defined as the following:  0,    nx − (5n − 1), kn (x) = −nx + (5n + 1),    0, if if if if x < 5 − n1 ; 5 − n1 ≤ x ≤ 5; 5 < x ≤ 5 + n1 ; x > 5 + n1 . We see that kn converges pointwise to k on R (see Figure 2) where, k(x) =  0, if x 6= 5; 1, if x = 5. This shows that functions can still be Baire one if they are discontinuous at one point. One of the questions one might ask is the following, how many points of continuity does a Baire one function have to have? 4 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 x Figure 2. The function kn , n = 1, 2, 3, 4 More specifically, can a Baire one function be everywhere discontinuous? The answer is no, and the reason will be addressed later on. In fact, we will see that Baire one functions must have many points of continuity. However, suppose f : R → R has a finite number of jump discontinuities at x = x1 , x2 , . . . , xk . Is f Baire one? The answer is yes. Let x = x1 , x2 , . . . , xk be the finite number of discontinuities of f , where x1 < x2 < x3 < · · · < xk . We will show the case when k = 1. Given any positive integer n, we see that f is continuous on the interval (−∞, x1 − 1/n), so define fn (x) = f (x) on this interval. Likewise, fn (x) = f (x) is continuous on the interval (x1 + 1/n, x2 ), where x = x2 is the next point of discontinuity. We then define our function fn such that fn (x) is a linear function on the interval [x1 − 1/n, x1 ] connecting the points f (x1 − 1/n) and f (x1 ) together (in order to guarantee the continuity of the fn s) and likewise on the interval [x1 , x1 + 1/n] connecting the points f (x1 ) and f (x1 + 1/n) in a linear manner. As a result, fn converges to f as n approaches infinity. The same process can be repeated for the other points {x2 , x3 , ..., xk } where f is discontinuous. We have just shown that functions with finitely-many points of discontinuities are Baire one. BAIRE ONE FUNCTIONS 5 Another interesting property is as follows. Suppose f and g are Baire one functions, both mapping D ⊆ R into R. Since the sequences {fn } and {gn } converge pointwise to f and g respectively, it follows by a property of convergent sequences that {fn + gn } converge to f + g and hence f + g is Baire one. Likewise, the sequence {fn gn } converges to f g by a similar argument and f g is Baire one [4]. Finally, we note that all differentiable functions are Baire one. This should not be too surprising, since all differentiable questions are continuous, and all continuous functions are Baire one by definition. A proof of this fact is given in the next theorem. Theorem 1. Suppose f : R → R is differentiable. Then f 0 is Baire one. Proof. We will show that f 0 (x) = limn→∞ fn (x). From calculus, we know that the definition of the derivative of a function f is defined as f (x + h) − f (x) . h→0 h f 0 (x) = lim We want our expression of f 0 to be in the form of an infinite limit. Hence, the following is an equivalent statement. f (x + h) − f (x) h→0 h
f x + n1 − f (x) = lim 1 n→∞   n   1 = lim n f x + − f (x) . n→∞ n     1 Let fn (x) = n f x + n − f (x) , and note that fn is continuous f 0 (x) = lim for n = 1, 2, 3 . . .. We conclude that,     1 − f (x) f (x) = lim n f x + n→∞ n = lim fn (x). 0 n→∞  6 4. Bounds and Uniform Convergence In this section, we present some properties of Baire one functions that pertain to bounds. An interesting question one might ask is whether a Baire one function f : [a, b] → R must be bounded that is, does there exist a real number M such that |f (x)| ≤ M for all x ∈ [a, b]? We first give an example of an unbounded Baire one function f : [0, 1] → R. Next, we give two properties of bounded Baire one functions and conclude with a theorem about uniform convergence. Let fn and f be defined as follows on the closed interval [0, 1]. We note that {fn } is a sequence of continuous functions.  2 n x, if 0 ≤ x < n1 ; fn (x) = 1 if n1 ≤ x ≤ 1. , x  0, if x = 0; f (x) = 1/x, if 0 < x ≤ 1. It’s obvious that f is unbounded on [0, 1]. We will show that fn converges to f pointwise on [0, 1]. Let x ∈ [0, 1]. Then we have two cases to consider. If x = 0, we have fn (0) = 0 = f (0) for all n. If 0 < x ≤ 1, we note the following. Choose a positive integer N such that 1/N < x. Then fn (x) = 1/x = f (x) for n ≥ N , since 1/n ≤ 1/N < x. It follows that lim fn (x) = lim f (x) = f (x). n→∞ n→∞ Hence, fn converges pointwise to f . We have just found a Baire one function that is unbounded. The next result shows that given a Baire one function with bound M , we can find a sequence of continuous function {fn } that converges pointwise to f with the same bound. Lemma 1. If f : [a, b] → R is a bounded Baire one function with bound M , then there is a sequence {fn } of continuous functions such that each fn has the same bound M and such that {fn } converges pointwise to f on [a, b]. Proof. Let {gn } be a sequence of continuous functions that converges pointwise to f on a closed interval [a, b]. Suppose also that f has bound M . Let {fn } be the sequence defined piecewise by BAIRE ONE FUNCTIONS 7   −M, if gn (x) < −M ; gn (x), if −M ≤ gn (x) ≤ M ; fn (x) =  M, if gn (x) > M . It is apparent that each fn has bound M . Since f also has bound M and each gn converges pointwise to f on [a, b], it follows that {fn } converges pointwise to f on [a, b] with bound M .  We use this result in the proof of the following lemma, which will in turn be used to help show the relationship between Baire one and uniform convergence. Lemma 2. P Let {fk } be a sequence of Baire one functions defined on [a, b] and let ∞ k=1 Mk be a convergent sequence of positive real numbers. If |fk (x)| ≤ M k for all k and for all x ∈ [a, b], then the function P∞ f (x) = k=1 fk (x) is a Baire one function. Proof. Let P∞{fk } be a sequence of Baire one functions defined on [a, b] and let k=1 Mk be a convergent sequence of positive real numbers.Since each fk is Baire one, for each positive integer k, there exists a sequence {gki } of continuous functions such that gki converges pointwise to fk on [a, b] and by Lemma 1, |gki | ≤ Mk for all i. For each positive integer n, let hn = g1n + g2n + · · · + gnn . We will show that {hn } converges pointwise to f on [a, b]. Fix a point x ∈ [a, b] and P let  > 0. By assumption, there exists a positive integer K such that ∞ k=K+1 Mk < . Choose an integer N > K such that |gki (x) − fk (x)| < K for 1 ≤ k ≤ K and for i ≥ N . For any n ≥ N we have, |hn (x) − f (x)| = ≤ n X k=1 n X gnk (x) − ∞ X fk (x) k=1 [gkn (x) − fk (x)] + K X |gkn (x) − fk (x)| + k=1 < K X k=1 < 3 fk (x) k=n+1 k=1 ≤ ∞ X n X k=K+1 ∞ X  +2 Mk K k=K+1 |gkn (x)| + ∞ X k=K+1 |fk (x)| 8 Since  > 0 was arbitrary, it follows that {hn (x)} converges pointwise to f (x) for all x ∈ [a, b].  While the definition of Baire category one functions relies simply on the fact that a sequence of continuous functions converges pointwise to a limit function f , pointwise convergence does not guarantee that the limit function necessarily retains the same properties of each of the functions fn . In order to ensure that the functions fn and f have the same properties, a stronger kind of convergence (defined below) is required [4]. Definition 3 (Uniform Convergence). Let {fn } be a sequence of functions defined on a set D and let f be defined on D. The sequence {fn } converges uniformly to f on D if for each  > 0 there exists a positive integer N such that |fn (x) − f (x)| <  for all x ∈ D and for all n ≥ N . From the definition we see that uniform convergence is pointwise convergence with the added property that given any  > 0, the choice of N is independent of x ∈ D [4]. Figure 3 shows that for sufficiently large N , the graph of fn must lie between the dotted lines around the graph of f . To give a simple example that illustrates the definition, consider the sequence {fn } defined by fn (x) = cos(nx)/n for all n. We will show that this sequence converges uniformly to the function f defined by f (x) = 0 for all x ∈ R. Let  > 0 and choose a positive integer N such that N > 1/. Then for all n ≥ N and x ∈ R, |fn (x) − f (x)| = cos(nx) 1 1 ≤ ≤ < . n n N The inequality shows that the sequence of functions {fn } converges uniformly to the limit function f on R. Our goal is to show that the limit function f of a uniformly convergent sequence of Baire one functions {fn } is also Baire one. Theorem 2. Let {fn } be a sequence of Baire one functions defined on [a, b]. If {fn } converges uniformly to f on [a, b], then f is a Baire one function. Proof. Let {fn } be a sequence of Baire one functions that converges uniformly to f on [a, b]. By definition of uniform convergence, for each k, there exists a subsequence {fnk } such that |fnk (x) − f (x)| < 2−k for all x ∈ [a, b]. BAIRE ONE FUNCTIONS 9 Figure 3. Illustration of Uniform Convergence [4] Let’s consider the sequence {fnk+1 −fnk }. We see that by the triangle inequality, |fnk+1 (x) − fnk (x)| ≤ |fnk+1 (x) − f (x)| + |f (x) − fnk (x)| ≤ 2−(k+1) + 2−k   1 −k 3 −k −k = (2 ) + 2 = 2 . 2 2 −k Let and note that |fnk+1 (x) − fnk (x)| ≤ Mk and P∞ Mk = (3/2)2 k=1 PMk is a convergent geometric series. We will show that the function ∞ k=1 [fnk+1 (x) − fnk (x)] is Baire one and use the result of Lemma 2 to show that f is Baire one. Thus, ∞ X [fnk+1 (x) − fnk (x)] = k=1 = lim N →∞ lim N →∞ N X [fnk+1 (x) − fnk (x)] k=1 N X fnk+1 (x) − k=1 N X fnk (x) k=1 = f (x) − fn1 (x). By hypothesis, theP function fn1 (x) is Baire one. It follows by Lemma 2 that the function ∞ k=1 [fnk+1 (x) − fnk (x)] = f (x) − fn1 (x) is Baire one. Since f (x) = (f (x) − fn1 (x)) + fn1 (x) and the sum of two Baire one functions is Baire one, f is also Baire one.  5. Fσ Sets We now try to find a way to determine whether or not a function is Baire one by using properties and results of Fσ sets. We first define 10 several properties of Fσ sets. Then, we digress briefly to present a theorem about continuous functions that is an analog of our main result of this section. Finally, we present and prove our main result. Definition 4 (Fσ set). [5] A subset A ⊆ R is called an S Fσ set if A is a countable union of closed sets. In other words, A = ∞ n=1 Fn where each Fn is a closed subset of R. The following are examples of Fσ sets on the real line. Let X = (0, 1), Y = [0, 1), and Z = [0, 1] respectively. Then we have,  ∞  [ 1 1 ; X = ,1− n n n=3  ∞  [ 1 ; Y = 0, 1 − n n=2 Z = ∞ [ [0, 1]; n=1 which are all countable unions of closed sets of R. It’s obvious from the definition that every closed set is also an Fσ set, but our constructions of X and Y show there exist sets that are Fσ and not closed. In fact, all open subsets of R are also Fσ [4]. We now present some properties of Fσ sets. Lemma 3. The intersection of two Fσ sets is an Fσ set. S S∞ Proof. Let A = ∞ i=1 Ai and B = j=1 Bj , such that Ai ’s and Bj ’s are closed sets. We see that by DeMorgan’s Laws, ∞ ∞ ∞ A ∩ B = (∪i=1 Ai ) ∩ (∪∞ j=1 Bj ) = ∪i=1 ∪j=1 (Ai ∩ Bj ) For each i and j, the set Ai ∩ Bj is a closed set. Hence A ∩ B is an Fσ set.  Lemma 4. Suppose that K and E are closed subsets of R. Then K −E is an Fσ set. Proof. Let K and E be closed sets. Since every closed set is an Fσ set and every open subset of R is also an Fσ set [4], it follows from Lemma 3 that, K − E = K ∩ EC is an Fσ set, where E C denotes the complement of the set E (which is an open set).  BAIRE ONE FUNCTIONS 11 The next result states that a closed interval [a, b] which equals a countable union of Fσ sets, can also be written as a countable union of “smaller” disjoint Fσ sets. It will be used in the proof of the main theorem of this section which gives a criterion that determines whether or not a function is Baire one. S LemmaS5. Suppose [a, b] = nk=1 Ak and each Ak is an Fσ set. Then [a, b] = nk=1 Bk where each Bk is an Fσ set, Bk ⊆ Ak for each k, and the Bk ’s are pairwise disjoint. S Proof. Suppose [a, b] = nk=1 Ak and each Ak is an Fσ set. By definition of Fσ sets, each S set Ak isSa countable union of closed sets. It follows that [a, b] = nk=1 Ak = ∞ i=1 Ei where each Ei is closed and Ei ⊆ Ak for some k. Next, let H1 = E1 , and Hi = Ei − (E1 ∪ · · · ∪ Ei−1 ) for i S ≥ 2. It follows from Lemma 4 that each Hi is an S Fσ set. SinceS[a, b] = nk=1 Ak and each Hi ⊆ Ak for some k, [a, b] = nk=1 Ak = ∞ i=1 Hi . We will now show that each of the Hi ’s are pairwise disjoint. We see that by DeMorgan’s Laws, H1 ∩ H2 = E1 ∩ (E2 − E1 ) = ∅ H2 ∩ H3 = (E2 − E1 ) ∩ [E3 − (E1 ∪ E2 )] = (E2 − E1 ) ∩ [(E3 − E1 ) ∩ (E3 − E2 )] = ∅ .. . Hi ∩ Hi−1 = [Ei − (E1 ∪ · · · ∪ Ei−1 )] ∩ [Ei−1 − (E1 ∪ · · · ∪ Ei−2 )] = (Ei − E1 ) ∩ [Ei − (E2 ∪ · · · ∪ Ei−1 )] ∩ (Ei−1 − E1 ) ∩ [Ei−1 − (E2 ∪ · · · ∪ Ei−2 )] = ∅. Hence, the Hi ’s are pairwise disjoint. Now, define Π1 = {i : Hi ⊆ A1 } and Πk = {i : Z+ − (Π1 ∪ · · · ∪ Πk−1 ) : Hi ⊆ Ak }, for 2 ≤ k ≤ n. Also, let [ Bk = Hi . i∈Πk We will show that the Bk ’s are pairwise disjoint Fσ sets, Bk ⊆ Ak for Sn each k, and conclude that [a, b] = k=1 Bk . Since each Bk is made up of pairwise disjoint Fσ sets Hi , it follows that the Bk ’s are pairwise disjoint Fσ sets as well. Also, since each Hi ⊆ Ak for some k, it follows that Bk ⊆ Ak for all k. Lastly, since 12 S [a, b] = ∞ i=1 Hi , we ask if all of the Hi ’s are used to build each of the Bk ’s. The answer is yes, since by definition of Bk , for a givenSj, Hj is used to build Bl where l = min{k : Hj ⊆ Ak }. Hence, [a, b] = nk=1 Bk . This completes the proof.  We digress briefly to present a theorem concerning continuous functions. This will “motivate” an analogous result for Baire one functions. We first present a definition and several results that will be used in the proof of Theorem 3. Definition 5. [4] Let A be a set of real numbers. (1) A set E ⊆ A is open in A if there is an open set O such that E = A ∩ O. (2) A set E ⊆ A is closed in A if there is a closed set F such that E = A ∩ F. Lemma 6. [4] Let A be a set of real numbers. (1) A set E ⊆ A is open in A if and only if A − E is closed in A. (2) A set E ⊆ A is closed in A if and only if A − E is open in A. (3) If A is an open set, then a set E ⊆ A is open in A if and only if it is open. (4) If A is an closed set, then a set E ⊆ A is closed in A if and only if it is closed. Lemma 7. [4] Let E be a set of real numbers. A point x is a limit point of E if and only if there exists a sequence of points in E − {x} that converges to x. We are now ready to prove the following result. Theorem 3. A function f : [a, b] → R is continuous on [a, b] if and only if the sets {x ∈ [a, b] : f (x) < r} and {x ∈ [a, b] : f (x) > r} are open in [a, b] for each real number r. Proof. Suppose that the sets {x ∈ [a, b] : f (x) < r} and {x ∈ [a, b] : f (x) > r} are open in [a, b] for each real number r. Let c ∈ [a, b] and let  > 0. By definition, we see that there exist open sets O1 and O2 such that {x ∈ [a, b] : f (x) < f (c) + } = [a, b] ∩ O1 {x ∈ [a, b] : f (x) > f (c) − } = [a, b] ∩ O2 where r = f (c) ± . Since c ∈ [a, b], and O1 and O2 are open sets, the set O = O1 ∩ O2 is an open set that contains the point c. Next, choose δ > 0 such that BAIRE ONE FUNCTIONS 13 the interval (c − δ, c + δ) ⊆ O. If |x − c| < δ and x ∈ [a, b], then x ∈ [a, b] ∩ O1 and x ∈ [a, b] ∩ O2 by the definition of O. Hence, f (c) −  < f (x) < f (c) + . It follows that |f (x) − f (c)| <  for all x ∈ [a, b] such that |x − c| < δ. This shows that f is continuous on [a, b]. Next, suppose that f is continuous on [a, b] and let r be a real number. To show that the sets {x ∈ [a, b] : f (x) < r} and {x ∈ [a, b] : f (x) > r} are open in [a, b] for each real number r, we will use properties (2) and (4) of Lemma 6, and Lemma 7 to show that the sets {x ∈ [a, b] : f (x) ≤ r} {x ∈ [a, b] : f (x) ≥ r} are closed in [a, b], that is each set contains all of its limit points. First, let E = {x ∈ [a, b] : f (x) ≥ r} and suppose w is a limit point of E, that is, for each positive number m, the set (w − m, w + m) ∩ E contains a point of E other than w. By Lemma 7, there exists a sequence {xn } in E − {w} that converges to w. By assumption, f is continuous at w, so the sequence {f (xn )} converges to f (w). Since f (xn ) ≥ r for all n, we have f (w) ≥ r. Thus, w is in E. Likewise, let G = {x ∈ [a, b] : f (x) ≤ r} and suppose d is a limit point of G. By Lemma 7, there exists a sequence {yn } in G − {d} that converges to d. By assumption f is continuous at d, so the sequence {f (yn )} converges to f (d). Since f (yn ) ≤ r for all n, we have f (d) ≤ r. Thus, d is in G. We have thus shown that E and G each contain all of their limit points. By definition the sets E and G are closed.  We now introduce the concept of the characteristic function of a set B ⊆ R. Lemma 8 is called the Tietze Extension Theorem, which is a result from analysis (or, more generally, topology) [4]. Lemma 9 is a result that uses both the definition of the characteristic function of a set and the Tietze Extension Theorem, and will also be used in the final theorem of this section. Definition 6. Let B ⊆ R. The characteristic function of B is the real-valued function χB : R → R defined by the equation  1, if x ∈ B; χB (x) = 0, if x ∈ R − B. 14 Lemma 8 (Tietze Extension Theorem). Let E be a closed set. If a function f : E → R is continuous on E, then there exists a continuous function fe : R → R such that f (x) = fe (x) for all x ∈ E. S Lemma 9. Suppose that [a, b] = nk=1 Bk where each Bk is an Fσ set and the Bk ’s are pairwise disjoint. Let n X f (x) = ck χBk (x) k=1 for x ∈ [a, b]. Then f is a Baire one function. S k k Proof. For each k let Bk = ∞ i=1 Ei where the sequence {Ei } is a nondecreasing sequence of closed sets. This is possible because by assumption, each Bk is an Fσ set. For each positive integer m, let n X fm = c k χE m k . k=1 S k is We will verify that the restriction of fm to the closed set nk=1 Em continuous. S k . Suppose there exists a sequence {xl } that converges to x0 ∈ nk=1 Em j Let x0 ∈ Em for some fixed j, 1 ≤ j ≤ n. We claim that all but j finitely many of the xl ’s are in Em . Suppose not, then there exists a k k subsequence {xln } ⊆ Em , such that {xln } converges to x0 ∈ Em . Since k j each Em is closed and disjoint, the set Em contains finitely many points xl . Thus, for sufficiently large l, f (xl ) = cj = f (x0 ). This certainly implies that f (xl ) converges Sn to fk (x0 ). By definition, f is continuous at x0 and hence on all of k=1 Em . S k Now let gm be the linear extension of fm from nk=1 Em to all of R. By the Tietze Extension Theorem gm is continuous on R. Hence, gm is continuous on [a, b]. Finally, since f (x) = lim gm (x) m→∞ for all x ∈ [a, b], the sequence {gm } converges pointwise to f on [a, b]. It follows that f is the pointwise limit of a sequence of continuous functions. By definition, f is Baire one.  We now present the main result of this section, and as stated earlier, it “looks” very similar to Theorem 3 concerning continuous functions. It follows from the next theorem that a function f is Baire one if and only if two sets involving the inverse image of f are Fσ sets. BAIRE ONE FUNCTIONS 15 Theorem 4. A function f : [a, b] → R is a Baire one function if and only if the sets {x ∈ [a, b] : f (x) < r} and {x ∈ [a, b] : f (x) > r} are Fσ sets for each real number r. Proof. Suppose that f is a Baire one function. Let {fn } be a sequence of continuous functions that converges pointwise to f on [a, b], and let r be a real number. It can be shown that {x ∈ [a, b] : f (x) < r} = ∞ [ ∞  \ ∞  [ k=1 p=1 n=p 1 x ∈ [a, b] : fn (x) ≤ r − k  . Since fn is a sequence of continuous functions, the set   1 x ∈ [a, b] : fn (x) ≤ r − k is closed, since the inverse image of a closed set is closed [4]. It follows that the set {x ∈ [a, b] : f (x) < r} is an Fσ set since the countable union of intersections of closed sets is an Fσ set. Since f is Baire one by assumption, −f is Baire one by properties of Baire one functions. Hence, the set {x ∈ [a, b] : f (x) > r} is also an Fσ set using a similar argument. For the converse, suppose that f is a bounded function such that the sets {x ∈ [a, b] : f (x) < r} and {x ∈ [a, b] : f (x) > r} are Fσ sets for each real number r. We will show the general case (when f is unbounded) later on. We use the following construction to produce a sequence {fn } of Baire one functions. Choose M > 0 such that |f (x)| < M for all x ∈ [a, b]. Next, fix n and let yk = −M + 2M k , n for 0 ≤ k ≤ n. Also, let Ak = {x ∈ [a, b] : yk−1 < f (x) < yk+1 }, for 1 ≤ k ≤ n. Let A = {x ∈ [a, b] : f (x) < yk+1 } B = {x ∈ [a, b] : f (x) ≤ yk−1 }. We will show that Ak = A ∩ B C is an Fσ set, where B C denotes the complement of B. By hypothesis, A is an Fσ set. We see that B C = {x ∈ [a, b] : f (x) > yk−1 } is an Fσ set by the same argument. 16 Since the intersection of two Fσ sets is also an Fσ set, it follows that Ak is an Fσ set as well. Also by hypothesis, [a, b] = n−1 [ Ak . k =1 S By Lemma 5, [a, b] = n−1 k=1 Bk , where each Bk is an Fσ set, Bk ⊆ Ak for each k, and the Bk ’s are pairwise disjoint. For each n, define fn : [a, b] → R as n−1 X fn (x) = yk χBk (x). k=1 By Lemma 9, fn is a Baire one function. The sets B1 , B2 , ..., Bn−1 are pairwise disjoint and partition [a, b]. Next, fix n and x ∈ [a, b]. There exists j, 1 ≤ j ≤ n, such that x ∈ Bj and Bj ⊆ Aj . By definition, fn (x) = yj , for x ∈ Bj . We know that yj−1 < f (x) < yj+1 , Since fn (x) = yj , we have, yj−1 < yj < yj+1 . yj−1 < fn (x) < yj+1 . Thus, by definition of yk above, 1 |yj+1 − yj−1 | 2 2M = , n for all x ∈ [a, b] and all n ∈ N. By definition, the sequence {fn } converges uniformly to f on [a, b]. By Theorem 2, f is a Baire one function. |fn (x) − f (x)| < For the general case (where f is not necessarily bounded), let h : R → (0, 1) be a continuous increasing function. For each real number r we have, (h ◦ f )−1 (r, ∞) = f −1 (h−1 (r, ∞)) by properties of functions. We will show that,  if r ≤ 0;  [a, b], −1 −1 −1 f (h (r), ∞), if 0 < r < 1; (h ◦ f ) (r, ∞) =  ∅, if r ≥ 1, where f : [a, b] → R, h : R → (0, 1), and (h ◦ f ) : [a, b] → (0, 1). We note that h ◦ f is a bounded function. BAIRE ONE FUNCTIONS 17 For r ≤ 0, we have, h−1 ((r, ∞)) = {x ∈ R : h(x) ∈ (r, ∞)} = {x ∈ R : h(x) > r} = R. Thus, (h ◦ f )−1 (r, ∞) = f −1 (h−1 (r, ∞)) = f −1 (R) = [a, b]. For 0 < r < 1, we see that, h−1 ((r, ∞)) = = = = {x ∈ R : h(x) ∈ (r, ∞)} {x ∈ R : h(x) > r} {x ∈ R : x > h−1 (r)} (h−1 (r), ∞). Thus, (h ◦ f )−1 (r, ∞) = f −1 (h−1 (r, ∞)) = f −1 (h−1 (r), ∞). Finally, for r ≥ 1, h−1 ((r, ∞)) = {x ∈ R : h(x) ∈ (r, ∞)} = {x ∈ R : h(x) > r} = ∅ Thus, (h ◦ f )−1 (r, ∞) = f −1 (h−1 (r, ∞)) = f −1 (∅) = ∅. Since, (h ◦ f )−1 (r, ∞) = f −1 (h−1 (r, ∞)) = {x ∈ [a, b] : f (x) ∈ (h−1 (r), ∞)} = {x ∈ [a, b] : f (x) > h−1 (r)}, it follows that h ◦ f satisfies the condition in the theorem and hence is a Baire one function. By properties of functions, f = h−1 ◦ (h ◦ f ). Since h is continuous and increasing, it is Baire one. By properties of limits, the composition of a continuous and a Baire one function is also Baire one. It follows that f is a Baire one function. This completes the proof.  As an illustration of Theorem 4, consider the characteristic function χQ on the set Q of rational numbers,  1, if x ∈ Q; χQ (x) = 0, if x ∈ R − Q. 18 We know that the set of irrational numbers R−Q is not an Fσ set [4]. Let r = 1. The set {x ∈ [a, b] : χQ < 1} = {x ∈ [a, b] : χQ = 0} = R−Q is not an Fσ set. Hence, the function χQ is not a Baire one function.1 6. Points of Continuity of f Recall that many Baire one functions are discontinuous, but certainly not all discontinuous functions are necessarily Baire one. How discontinuous are Baire one functions? Since the nowhere continuous function χQ is not a Baire one function, we may wonder whether Baire one functions must be continuous somewhere over its domain. In fact, we will show that if f is a Baire one function, then every closed interval contains a point of continuity of f . We need some new definitions to help us with our proof. In addition, the following lemmas will be useful. Let f : R → R be a fixed function. Definition 7. Let A ⊆ R. The extended (possibly infinite) real number ω(A) = sup{|f (x) − f (y)| : x, y ∈ A} is called the oscillation of f on A. One can think of the oscillation of a function f on a set A as roughly the largest distance between the maximum and minimum values of f (x) for all x ∈ A. We can also define the oscillation of a function f at a point. This is shown as follows. Definition 8. For x0 ∈ R, let Aδ = (x0 − δ, x0 + δ) for each δ > 0. The oscillation of f at x0 is the number ω(x0 ) = limδ→0 ω(Aδ ). Let’s use this definition to show a simple (yet important) result involving continuity. Lemma 10. f is continuous at x0 if and only if ω(x0 ) = 0. Proof. First, suppose f is continuous at x0 and let  > 0. By definition, there exists δ > 0 such that |f (x) − f (x0 )| < 3 . Then, for all x, y ∈ Aδ |f (x) − f (y)| ≤ |f (x) − f (x0 )| + |f (x0 ) − f (y)|   2 < + = < . 3 3 3 It follows that sup{|f (x) − f (y)| : x, y ∈ Aδ } ≤ 23  < . Since  > 0 was arbitrary, we have ω(x0 ) = 0. For the converse, suppose ω(x0 ) = 0 and let  > 0. There exists δ0 > 0 such that |ω(Aδ )| <  for all x such that |x − x0 | < δ0 . Thus 1While not Baire one, it can be shown that χQ is Baire class two. See [7]. BAIRE ONE FUNCTIONS 19 by definition, limδ→0 sup{|f (x) − f (y)| : x, y ∈ Aδ } = 0 and |f (x) − f (x0 )| < . Therefore, x is continuous at x0 .  The next lemma will be used in our proof of Theorem 5. It is mentioned in discussion notes [5], and it won’t be proved here. It is a version of a famous theorem called the Baire Category Theorem. Lemma 11. Suppose that {Dn } is a sequence of closed subsets of R S and that [a, b] = ∞ D n . Then at least one of the sets Dn contains a n=1 closed interval. We are now ready to prove the following. Theorem 5. If f : R → R is a Baire one function, then every closed interval contains a point of continuity of f . Proof. Let I0 be a closed interval and {fn } be a sequence of continuous functions on R such that limn→∞ fn (x) = f (x) for all x ∈ R. We will prove that I0 contains a point of continuity of f . Let  > 0. We first show that there exists a closed interval I1 ⊆ I0◦ such that ω(I1 ) ≤ , where I0◦ is the interior of I0 (or the non-endpoints of I0 ). For m, n ∈ N, let (1) Anm =    . x ∈ I0 : |fn (x) − fn+m (x)| ≤ 3 Since fn is a sequence of continuous functions and the inverse image of a closed set is closed, the set Anm is closed [4]. By properties of closed sets, the set ∞ \ Dn = Anm m=1 is closed for all n We will now show that ∞ [ I0 = Dn . n=1 We note that Dn ⊆ I0 by definition. Hence, we need to show that I0 ⊆ Dn for some n. In other words, given a fixed x ∈ I0 , we must find an N such that x ∈ AN m for all m. Let  > 0. Since by hypothesis, limn→∞ fn (x) = f (x) for all x ∈ R, there exists a positive integer N such that |fn (x) − f (x)| ≤ 6 for all n ≥ N . It follows that, 20 |fN (x) − fN +m (x)| ≤ |fN (x) − f (x)| + |f (x) − fN +m (x)|    ≤ + = . 6 6 3 Thus, x ∈ AN m for all m, x ∈ DN , and I0 ⊆ Dn for some n. By Lemma 11, there exists n ∈ N and a closed interval J such that J ⊆ Dn . Next, we note that for a fixed n, fn+m (x) converges pointwise to f (x) for all x ∈ I0 . It follows from (1) that  for all x ∈ J. 3 We now show that there exists a closed interval I1 ⊆ J ◦ ⊆ I0◦ such that ω(I1 , f ) ≤ . First, we show that ω(I1 , fn ) ≤ 3 . This new notation denotes the oscillation of fn over the interval I1 . Since each fn is continuous on J, then fn is uniformly continuous on J [4]. Let  > 0. There exists δ > 0 such that |fn (x) − fn (y)| ≤ 3 for all x, y ∈ J that satisfies |x − y| ≤ δ. Since I1 ⊆ J we have, |fn (x) − f (x)| ≤ (2)  ω(I1 , fn ) = sup{|fn (x) − fn (y)| : x, y ∈ I1 } ≤ . 3 Now let x, y ∈ I1 . Using equations (2) and (3) we have, (3) |f (x) − f (y)| ≤ |f (x) − fn (x)| + |fn (x) − fn (y)| + |fn (y) − f (y)|    ≤ + + 3 3 3 = . We see that, ω(I1 , f ) = sup{|f (x) − f (y)| : x, y ∈ I1 } ≤ . Using the construction above, we construct a sequence {Ik } of closed intervals such that Ik+1 ⊆ Ik◦ ⊆ I0◦ for k = 1, 2, ... and 1 (4) ω(Ik , f ) ≤ . k By the Nested Intervals Theorem [4], the set ∞ \ Ik k =1 contains at least one point x0 and x0 ∈ I0 . It follows from (4) that ω(x0 ) = 0. By Lemma 10, f is continuous at x0 . This completes the proof.  BAIRE ONE FUNCTIONS 21 7. Baire One and First-Return Recoverability A well known fact from analysis is that a continuous function f : [0, 1] → [0, 1] is uniquely determined by its values on any countable dense set in [0, 1] (we define what it means for a set to be dense later on) [9]. We extend this idea to Baire one functions and ask whether they too can be determined by its values on any countable set. Since not all Baire one functions are continuous, the answer is no, but there are certain countable dense sets that give us some nice results. We conclude with a characterization of Baire one functions using the notion of first return recoverability. First, let’s state several definitions. Definition 9 (Dense set). [8] A set D ⊆ X is dense in X if for any point x ∈ X, any neighborhood of x contains at least one point from D. Equivalently, D is dense in X if and only if D = X, where D is the closure of D. Underlying our characterization is the idea of what we call a trajectory. This is defined below. Definition 10. A trajectory is any sequence {xn } of distinct points in [0, 1], whose range is dense in [0, 1]. Any countable dense set J ⊆ [0, 1] is called a support set and any enumeration of D is also a trajectory. If the value of a continuous function f : [0, 1] → R at each point in a trajectory is known, then there is an algorithm that computes the value of the function at any point in [0, 1]. Definition 11. Let {xn } be a fixed trajectory and let y ∈ [0, 1]. Suppose ρ > 0 and let Bρ (y) = {x ∈ [0, 1] : |x − y| < ρ}. Also, let r(Bρ (y)) be the first element of the trajectory in Bρ (y). The first return route to y, denoted Ry = {yk }, is given by the recursive formula y1 = x0 , yk+1 =  r(B|y−yk | (y)), if y = 6 yk yk , if y = yk . The function f : [0, 1] → R is called first return recoverable with respect to {xn } if for each y ∈ [0, 1], lim f (yk ) = f (y), k→∞ and f is first return recoverable if there exists a trajectory {xn } such that f is first return recoverable with respect to {xn }. 22 It can be shown that a function f : [0, 1] → R is continuous if and only if it is first-return recoverable with respect to every trajectory. However, if we weaken this condition, we obtain the following theorem characterizing Baire class one functions. Before we present the theorem, we introduce the definition of a Gδ set. It will be used in the proof of our last theorem. Definition 12 (Gδ set). [4] A subset G ⊆ R is called a Gδ T set if G is a countable intersection of open sets. In other words, G = ∞ n=1 Gn where each Gn is an open subset of R. Note that by DeMorgan’s Laws, the complement of a Gδ set is Fσ [4]. Theorem 6. A function f : [0, 1] → R is first return recoverable if and only if it is of Baire class one. Proof. We will prove the forward direction. The other direction is given in [6], which we leave to the reader. Suppose f : [0, 1] → R is first return recoverable with respect to a trajectory {xn }. It suffices to show that for each real number r, the set A = {x ∈ [0, 1] : f (x) ≥ r} is a Gδ set. If we can prove this, then we know that −f is also first return recoverable and that {x ∈ [0, 1] : −f (x) ≥ −r} is a Gδ set, or equivalently {x ∈ [0, 1] : f (x) ≤ r} is a Gδ set. It follows from Theorem 4 that f is Baire one. Since f is first return recoverable with respect to a trajectory {xn }, f (x) = limk→∞ f (yk ) for all x ∈ [0, 1] where {yk } = Ry , as defined in the definition above. Fix r ∈ R. We assume the following. (1) For each positive integer k, let Ik = (r − k1 , ∞). (2) Let {xnj } be a subsequence of the trajectory {xn } consisting of those xn ∈ f −1 (Ik ). In other words, those xn that satisfy f (xn ) > r − k1 . (3) For each j ∈ N, let dj = min{ 12 |xnj − xi | : i < nj }, and let Oj = (xnj − dj , xnj + dj ). Let  ∞ [ ∞ \ Hk = Oj ∪ {xn1 , xn2 , . . . , xni−1 } . i=1 j=i From (1), (2), and (3) above, we have A= ∞ \ k =1 Hk . BAIRE ONE FUNCTIONS 23 We now show that for each fixed k, S Hk is Gδ . First, since the union of any collection of open sets is open, ∞ j=i Oj is an open set. Second, the set {xn1 , xn2 , . . . , xni−1 } is Gδ because it has a finite number of elements [4]. Hence, it is equal to a countable intersection S T∞ of open sets. Let G = ∞ O and the set {x , x , . . . , x } = n1 n2 ni−1 j=i j n=1 Gn . It follows that, \  ∞ Hk = G ∪ Gn n=1  ∞  \ = G ∪ Gn n=1 where G and each Gn are open sets. Thus, Hk is a countable intersection of Gδ sets and is a Gδ set. It follows easily that A is a Gδ set as well.  8. Conclusion While the definition of a Baire one function is fairly straightforward, this paper shows that some of its properties and examples certainly are not. In fact, most research papers dealing with this topic require graduate-level analysis and/or topology in order to fully appreciate (and understand) the results. Hence, many of the (perhaps) more interesting results are not presented here. With that said, there is great potential for further study of Baire functions once one becomes more mathematically mature. Therefore, this paper can be regarded as a starting point to a future, more in-depth project or thesis. References [1] MacTutor History of Mathematics. René-Louis Baire. http://www-groups. dcs.st-and.ac.uk/~history/Biographies/Baire.html. [2] F.A. Medvedev (trans. Roger Cooke). Scenes from the History of Real Functions, Birkhauser, 1992. [3] Wikipedia. Baire Functions. http://en.wikipedia.org/Baire. [4] R.A. Gordon, Real Analysis: A First Course, Addison-Wesley, Boston, 2nd ed., 2002. [5] R.A. Fontenot, Discussion notes [6] U.B. Darji, M.J. Evans, R.J. O’Malley, A First-Return Characterization of Baire 1 Functions, Real Analysis Exchange, 19(1993-94), 510-515. [7] E.R. Lorch, Continuity and Baire Functions, The American Mathematical Monthly, 78, No.7(1971), 748-762. [8] Wikipedia. Dense Set. http://en.wikipedia.org/wiki/Dense_set. 24 [9] M.J. Evans, R.J. O’Malley, First-Return Limiting Notions in Real Analysis, Real Analysis Exchange, 29(2003-04), 503-530.