Global singularization and the failure of SCH

We say that κ is µ-hypermeasurable (or µ-strong) for a cardinal µ ≥ κ + if there is an embedding j: V → M with critical point κ such that H(µ) V is included in M and j(κ)> µ. Such j is called a witnessing embedding. Building on the results in [7], 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(κ)-hypermeasurable cardinals κ, where F(κ)> κ +, with a witnessing embedding j such that either j(F)(κ) = κ + or j(F)(κ) ≥ F(κ), are turned into singular strong limit cardinals with cofinality ω. 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 κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2 α ∈ {α +, α ++} for every cardinal α below κ (in this case every κ ++-hypermeasurable cardinal in the ground model is witnessed either by a j with either j(F)(κ) ≥ F(κ) or j(F)(κ) = κ +)