Abstract
Let A be point separating unital subalgebra of C(T) where T is a compact metric space. For each bounded function f:T→R which is continuous on the complement of a meagre subset of T there exists a sequence (wn) of elements of the algebra A such that the sequence (wn) convergence uniformly to the function f on each compact subset of the interior of the continuity points of the function f.
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References
Bierstedt, K. D., Verallgemeinerungen des Satzes von Stone-Weierstrass, in: Jahrbuch Überblicke Mathematik (1975), 109–135.
Prolla, J.B., Weierstrass-Stone, the Theorem, Peter Lang Verlag, Frankfurt, 1993.
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Brosowski, B., da Silva, A.R. A Stone-Weierstrass theorem for certain discontinuous functions. Approx. Theory & its Appl. 13, 83–87 (1997). https://doi.org/10.1007/BF02836263
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DOI: https://doi.org/10.1007/BF02836263