414/514 Simple examples of Baire class one functions

October 6, 2014

Recall that a real-valued function f defined on an interval I is (in) Baire class one (\mathcal B_1) iff it is the pointwise limit of continuous functions.

Examples are continuous functions, of course, but functions in \mathcal B_1 do not need to be continuous. An easy example is the function f:[0,1]\to\mathbb R given by f(x)=0 if x\ne 1 and f(x)=1 if x=1. This is the pointwise limit of the functions f_n(x)=x^n. By the way, an easy modification of this example shows that any function that is zero except at finitely many points is in \mathcal B_1.

Step functions are another source of examples. Suppose that a=x_0<x_1<\dots<x_{n-1}<x_n=b and that s:[a,b]\to\mathbb R is constant on each (x_i,x_{i+1}). Then s is the pointwise limit of the functions s_k, defined as follows: Fix a decreasing sequence \epsilon_k converging to 0, with \epsilon_k\le 1/k and 2\epsilon_k<x_{i+1}-x_i for all i. Now define \hat s_k as the restriction of f to

\displaystyle \{x_0,x_1,\dots,x_n\}\cup\bigcup_{i=0}^{n-1}[x_i+\epsilon_k,x_{i+1}-\epsilon_k],

and let s_k:[a,b]\to\mathbb R extend \hat s_k 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 f:\mathbb R\to\mathbb R and f(x)=g'(x) for all x. This is the pointwise limit of the functions f_n(x) given by

f_n(x)=\displaystyle\frac{g\left(x+\frac1n\right)-g(x)}{\frac1n}.

This simple construction does not quite work if f is defined on a bounded interval (as x+1/n may fall outside the interval for some values of x). We can modify this easily by using straight segments as in the case of step functions: Say f:[a,b]\to\mathbb R. For n large enough so 1/n<b-a, define f_n(x) as above for x\in[a,b-1/n], and now set f_n(b)=f(b) and extend f_n linearly in the interval {}[b-1/n,b].

Additional examples can be obtained by observing, first, that \mathcal B_1 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 \mathcal B_1, since monotone functions are the uniform limit of step functions on bounded intervals: Given an increasing f:[a,b]\to\mathbb R, let f_n(x)=\lfloor nf(x)\rfloor/n. It follows that all functions of bounded variation are in \mathcal B_1, since any such function is the difference of two increasing functions.

Another interesting source of examples is characteristic functions. Given X\subseteq\mathbb R, the function \chi_X is in \mathcal B_1 iff X is both an \mathbf F_\sigma and a \mathbf G_\delta set.

On the other hand, \chi_{\mathbb Q} is not in \mathcal B_1, since it is discontinuous everywhere while Baire class one functions are continuous on a comeager set.