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An elementary example of a totally ordered nonBlumberg Baire space is given. A topological space X is called Blumberg if for each f: X-*R, there is a dense subset D of X such that f ID is continuous. Bradford and Goffman proved in [2] that a metric space X is Blumberg if and only if it is Baire. This result generalized the result of Blumberg in [1] that R is Blumberg. We give here an example of a totally ordered Baire space which fails to be Blumberg. A totally ordered set T is called an c%1-set if for any countable subsets A and B such that a that a supplied with the order topology. (Such an 1-set exists. See [3, p. 187].) Then X is a Baire space. For suppose {Gi})1 is a collection of open dense sets, and (a, b) is a nonempty basic open set in X. We claim (a, b) nfl 1 Gi$ 0. There is a nonempty open interval (a,, b,) c (a, b)nG,. Inductively, we find nonempty open intervals (an, bn) such that (a,+,, b,+l)c (an, b) rG+. Then aj xe X such that {ai}?=I proves that X is Baire. X is a P-space without isolated points (see [3, Problem 13.P]). Hence any dense subset of X is a P-space without isolated points. Therefore, if f :X-*R is a one-to-one function and D a dense subset of X, then f ID fails to be continuous at each point of D. Thus, X fails to be Blumberg. REFERENCES 1. H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), 113-128. 2. J. C. Bradford and C. Goffman, Metric spaces in which Blumberg's theorem holds, Proc. Amer. Math. Soc. 11 (1960), 667-670. MR 26 #3832. 3. L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #6994. DEPARTMENT OF MATHEMATICS, WASHINGTON UNIVERSITY, ST. Louis, MISSOURI 63130 Received by the editors February 2, 1973. AMS (MOS) subject classifications (1970). Primary 54C30, 54F05.

A totally ordered Baire space for which Blumberg’s theorem fails