We say that a subset X of **R**^{2}
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 **R**^{2}
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.**