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Baire Function


The class of continuous functions is called the Baire class 0. For each n, the functions that can be considered as pointwise limits of sequences of functions of Baire class n-1 but are not in any of the preceding classes are said to be of Baire class n. A Baire or (analytically representable) function is that function belonging to a Baire class n for some n. For example, discontinuous functions representable by Fourier series belong to class 1.

This notion was introduced by Baire in the 19th century. In 1905, Lebesgue showed that each of the Baire classes is nonempty and that there are (Lebesgue-) measurable functions that are not Baire functions (Kleiner 1989).


See also

Continuous Function

This entry contributed by Mohammad Sal Moslehian

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References

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 104-106, 1971.Kleiner, I. "Evolution of the Function Concept: A Brief Survey." College Math. J. 20, 282-300, 1989.

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Baire Function

Cite this as:

Moslehian, Mohammad Sal. "Baire Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BaireFunction.html

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