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 2α ∈ {α+, α++} for every cardinal α below κ (in this case every κ++-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F )(κ) ≥ F (κ) or j(F )(κ) = κ+).