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Rosenthal in [11] proved that if $\left(f_k\right)$ is a uniformly bounded sequence of real-valued functions which has no pointwise converging subsequence then $\left(f_k\right)$ has a subsequence which is equivalent to the unit basis of $l^1$ in the supremum norm. Kechris and Louveau in [6] classified the pointwise convergent sequences of continuous real-valued functions, which are defined on a compact metric space, by the aid of a countable ordinal index “$\gamma$”. In this paper we prove some local analogues of the above Rosenthal ’s theorem...

Kechris and Louveau in [5] classified the bounded Baire-1 functions, which are defined on a compact metric space $K$, to the subclasses ${ℬ}_1^{\xi }\left(K\right)$, $\xi <{\omega }_1$. In [8], for every ordinal $\xi <{\omega }_1$ we define a new type of convergence for sequences of real-valued functions ($\xi$-uniformly pointwise) which is between uniform and pointwise convergence. In this paper using this type of convergence we obtain a classification of pointwise convergent sequences of continuous real-valued functions defined on a compact metric space $K$, and...