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An equivalent definition of functions of the first Baire class. (English) Zbl 0970.26004
Let $$f:X \to Y$$ be a mapping between complete separable metric spaces $$(X,d_X)$$ and $$(Y,d_Y)$$. Then $$f$$ is of the first Baire class if and only if for each $$r > 0$$ there is a function $$\eta > 0$$ on $$X$$ such that $$d_Y(f(x),f(y)) < r$$ whenever $$d_X(x,y) < \min(\eta (x),\eta (y))$$.

##### MSC:
 26A21 Classification of real functions; Baire classification of sets and functions 54E50 Complete metric spaces 54C08 Weak and generalized continuity 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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##### References:
 [1] Baire, R., Sur les fonctions des variables réeles, Ann. Mat. Pura ed Appl. 3(1899), 1-122. [2] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901 [3] I. P. Natanson, Theory of functions of a real variable. Vol. II, Translated from the Russian by Leo F. Boron, Frederick Ungar Publishing Co., New York, 1961.
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