414/514 Simple examples of Baire class one functions

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.

This entry was posted on Monday, October 6th, 2014 at 10:19 am and is filed under 414/514: Analysis I. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

6 Responses to 414/514 Simple examples of Baire class one functions

andrescaicedo says:

October 6, 2014 at 10:23 am

Thanks to Stuart Nygard for suggesting the much easier argument for derivatives being $\mathcal B_1$ than the messier approach I suggested in lecture.

(Proofs of the closure of $\mathcal B_1$ under uniform limits, of the continuity fact, and of the claim about characteristic functions, will be provided in lecture.)

Reply
Monica says:

October 7, 2014 at 10:17 am

Sorry I am a bit confused .. Is this saying that we are setting f_n(b) = f(b) = g'(b)? So by taking it as the derivative, it approaches b but never passes it, or touches it?

Reply
- andrescaicedo says:
  
  October 7, 2014 at 12:07 pm
  
  Hi Monica.
  
  By construction, each $f_n$ is continuous.
  
  For any $\epsilon>0$ , if $n$ is large enough then $1/n<\epsilon$ . If $x<b$ , letting $\epsilon=b-x$ , we see that for all $n$ sufficiently large, $\displaystyle f_n(x)=\frac{g\left(x+\frac1n\right)-g(x)}{\frac1n}$ , and this expression converges to $g'(x)=f(x)$ .
  
  So the only issue with this definition is whether we also have $f_n(b)\to f(b)$ , but we arrange that this happens trivially, by setting $f_n(x)=f(b)$ for all $n$ .
  
  Putting all this together, we see that $f_n\to f$ pointwise.
  
  Reply
Monica says:

October 10, 2014 at 7:04 am

oh okay thanks Andres!

Reply
414/514 Examples of Baire class two functions | A kind of library says:

November 3, 2014 at 10:32 am

[…] Previously, we listed some examples of Baire class one functions. Here we do the same for functions in the next class of Baire. Recall that if is an interval, the function is (in) Baire class two () iff it is the pointwise limit of a sequence of Baire one functions. […]

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Neugebauer’s theorem | A kind of library says:

December 30, 2014 at 12:44 am

[…] derivatives are Darboux continuous (that is, they satisfy the intermediate value property), and are Baire one functions (that is, they are the pointwise limit of a sequence of continuous functions). But this […]

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