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It's quite easy to find functions of Baire classes 0 and 1; however, the regularity of these functions falls off quite quickly, and with it, ease of construction. The most famous example of a Baire class 2 function is the characteristic function of the rationals, which isn't even Riemann integrable. Other constructible Baire class 2 functions are given elsewhere on Stack Exchange.

What are some examples of Baire functions which aren't of classes 0, 1, or 2?

Aidan Backus
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The characteristic function $\chi_X$ of any Borel set $X$ is always a Baire function, and the higher the Borel rank of $X$ the higher the Baire rank of $\chi_X$ - so if you know how to construct Borel sets of high rank, you can construct Baire functions of high rank. For instance, the set of normal real numbers is Borel of rank $3$ (it's $\Pi^0_3$).

Noah Schweber
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  • I should've known to look at the Borel hierarchy. Thanks! – Aidan Backus Nov 11 '17 at 04:57
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    To expand a bit, the Borel measurable functions are precisely the functions such that the preimage of any Borel set is Borel. The Baire functions are the functions in some Baire class. The key theorem is that they coincide, and there is an appropriate level by level stratification which includes Noah's result, namely, a function is Baire class $\xi$ if and only if the preimage of any open set is $\mathbf{\Sigma}^0_{\xi+1}$. That said, much can be said beyond this observation, and specific examples are desirable. – Andrés E. Caicedo Nov 11 '17 at 05:03
  • @AndrésE.Caicedo, I need to one more Baire classes . Could you please refer me to a references as starting point. – 00GB May 27 '21 at 13:42
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    @00GB Books in descriptive set theory may cover the subject, in particular, Kechris's "Classical descriptive set theory". I also like "A second course on real functions", by van Rooij and Schikhof. – Andrés E. Caicedo May 28 '21 at 05:27
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    @AndrésE.Caicedo, Thank you so much. – 00GB May 28 '21 at 21:00