Building on the results in \cite{RADEKeaston}, we will show that if $V$ satisfies
GCH and $F$ is an Easton function from the regular cardinals into cardinals satisfying some
mild restrictions, then there exists a cardinal-preserving forcing extension $V^*$
where $F$ is realised on all $V$-regular cardinals and moreover: all
$F(\kappa)$-hypermeasurable cardinals $\kappa$, where $F(\kappa) > \kappa^+$, with a
witnessing embedding $j$ such that either $j(F)(\kappa) = \kappa^+$ or
$j(F)(\kappa) \ge F(\kappa)$, are turned into singular strong limit cardinals with
cofinality $\omega$. This provides some partial information about the possible structure
of a continuum function with respect to singular cardinals with countable cofinality.
As a corollary, this shows that the continuum function on a singular strong limit
cardinal $\kappa$ of cofinality $\omega$ is virtually independent of the behaviour
of the continuum function below $\kappa$, at least for continuum functions which
are simple in that $2^\alpha \in \{\alpha^+, \alpha^{++}\}$ for every cardinal
$\alpha$ below $\kappa$ (in this case every $\kappa^{++}$-hypermeasurable cardinal
in the ground model is witnessed either by a $j$ with either $j(F)(\kappa) \ge F(\kappa)$
or $j(F)(\kappa) = \kappa^+$).
Keywords: Easton's theorem, Prikry type forcings, hypermeasurable and strong cardinals, lifting