Easton's theorem does not prescribe singular-cardinal powers

Easton's realization theorem (Easton's theorem for regular cardinals) is a statement about the continuum function at regular cardinals: its conditions $\kappa <2^{\kappa }$, monotonicity and $\mathrm{cf}(2^{\kappa })>\kappa$ (Necessary constraints on the regular-cardinal continuum function) are exactly the constraints that the construction can meet there. It makes no assignment of $2^{\kappa }$ for singular $\kappa$ and gives no licence to read one off from a prescribed behaviour on regular cardinals.

At a singular cardinal $\kappa$ the same general constraints remain in force for the value: monotonicity gives $2^{\kappa }\ge 2^{\mu }$ for every $\mu <\kappa$, Cantor's theorem gives $\kappa <2^{\kappa }$, and König's theorem gives $\mathrm{cf}(2^{\kappa })>\kappa$ (Necessary constraints on the regular-cardinal continuum function, Assuming the Axiom of Choice: $\kappa <{\kappa }^{\mathrm{cf}(\kappa )}$ for every infinite cardinal $\kappa$, and $\mathrm{cf}(2^{\kappa })>\kappa$; in particular $\mathrm{cf}(2^{{\mathrm{\aleph }}_0})>{\mathrm{\aleph }}_0$). In the GCH case $2^{<\kappa }=\kappa$, so Cantor's strict inequality gives $2^{\kappa }\ge (2^{<\kappa }{)}^{+}$. The values at singular cardinals are governed by further theorems not proved on this page, and the page states nothing about them: in particular it does not claim that an arbitrary prescription on regular cardinals extends to a singular cardinal, and it does not claim the Singular Cardinal Hypothesis or its failure.

The Easton function of Easton functions on regular cardinals is therefore used only on its class of infinite regular cardinals, and the class-generic construction of Easton's theorem for regular cardinals is only asserted to realize the prescription there.