  We study a fine hierarchy of Borel-piecewise continuous functions,
especially, between closed-piecewise continuity and $G_\delta$-piecewise
continuity. Our aim is to understand how a priority argument in computability
theory is connected to the notion of $G_\delta$-piecewise continuity, and then
we utilize this connection to obtain separation results on subclasses of
$G_\delta$-piecewise continuous reductions for uniformization problems on
set-valued functions with compact graphs. This method is also applicable for
separating various non-constructive principles in the Weihrauch lattice.
