Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So:


1. What are the non-trivial constraints on continuum function in singular cardinals?
2. Is it possible to well-define a concept of "**Continuum function maximum of singular cardinals**" by these constraints?
3. How strong is the consistency strength of this definition?



Continuum function maximum
The intricacies of arithmetic at singular cardinals notwithstanding, I think you're looking for something which doesn't exist.


Given any cardinals $\kappa,\lambda$, singular or regular, there is a (set) forcing extension preserving cardinals and cofinalities in which $2^\kappa>\lambda$. Upper bounds only appear in a meaningful sense when we try to control the value of the continuum function on *many* inputs simultaneously. For example, famously Shelah proved that $2^{\aleph\_\omega}<\aleph\_{\omega\_4}$ **provided that** $2^{\aleph\_n}<\aleph\_\omega$ for all $n<\omega$, but it's perfectly consistent that $2^{\aleph\_\omega}\ge\aleph\_{\omega\_4}$ since for example we could have $2^{\aleph\_0}=\aleph\_{\omega\_{17}+42}$ already.


