We say that a subset X of R² is Sierpinski-Zygmund (shortly SZ-set) if it does not contain a partial continuous function of cardinality continuum c. We observe that the family of all such sets is cf(c)-additive ideal. Some examples of such sets are given. We also consider SZ-shiftable sets, that is, subsets X of R² for which there exists a function f:R-->R such that f+X is an SZ-set. Some results are proved about SZ-shiftable sets. In particular, we show that the union of two SZ-shiftable sets does not have to be SZ-shiftable.

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Last modified June 29, 2001.