## 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 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 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, 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.