We prove that the Covering Property Axiom CPA_{prism}^{game},
which holds in the iterated perfect set model, implies
that there exists an additive discontinuous
almost continuous function f from **R** to **R** whose graph
is of measure zero. We also show that, under CPA_{prism}^{game},
there exists a Hamel basis H for which the set
E^{+}(H), of all linear combinations of elements from
H with positive rational coefficients,
is of measure zero.
The existence of both of these examples follows from
Martin's axiom, while it is unknown whether
either of them can be constructed in ZFC.
As a tool for the constructions we will show that
CPA_{prism}^{game} implies its seemingly stronger version,
in which \omega_{1}-many games are played simultaneously.

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**Last modified February 2, 2006.**