This note shows that if a subset S of **R** is such that some continuous function f from **R** to **R** has the
property "f[S] contains a perfect set," then some infinitely many times differentiable function g (from **R** to **R**) has the same property.
Moreover, if f[S] is nowhere dense, then the g can have the stronger property "g[S] is perfect."
The last result is used to show that it is consistent with ZFC (the usual axioms of set theory) that
for each subset S of **R** of cardinality continuum there exists an infinitely many times differentiable
function g from **R** to **R** such that g[S] contains a perfect set.

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**Last modified September 27, 2012.**