Small coverings with smooth functions
under the Covering Property Axiom
by
Krzysztof Ciesielski,
and J. Pawlikowski
Canad. J. Math. 57(3) (2005), 471493.
In the paper we formulate a Covering Property Axiom CPA,
which holds in the iterated perfect set model,
and show that it implies the following facts.

There exists a family F of less then continuum many
C^{1} functions from R to R
such that R^{2} is covered by
functions from F
in the sense that
for every (x,y) in R^{2} there exists an
f in F such that either f(x)=y or f(y)=x.
 For every Borel function f:R>R there exists
a family F of less than continuum many "C^{1}" functions
(i.e., differentiable functions
with continuous derivatives, where derivative can be infinite)
whose graphs cover the graph of f.

For every positive n and
a D^{n} function f:R>R there exists
a family F of less than continuum many C^{n} functions
whose graphs cover the graph of f.
We also provide the examples showing that
in the above properties the smotheness conditions are the best
possible.
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Last modified February 10, 2004.