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l], showed that for every real function/ on the real line R there is a dense set DER such that/ is continuous on D relative to D. The purpose of this paper is to characterize the metric spaces in which Blumberg's theorem holds. Let 5 be a metric space and E a subset of 5. A point xES is said to be of the second category relative to E if every sphere of center x contains a subset of E which is of the second category in S. Otherwise, x is said to be of the first category relative to E. S is saiddoi:10.1090/s0002-9939-1960-0146310-1 fatcat:gavwvw3tund7hnrlnoleple3mu