  We weaken the assumption of summable variations in a paper by Verbitskiy
\cite{verb} to a weaker condition, Berbee's condition, in order for a 1-block
factor (a single site renormalisation) of the full shift space on finitely many
symbols to have a $g$-measure with a continuous $g$-function. But we also prove
by means of a counterexample, that this condition is (within constants)
optimal. The counterexample is based on the second of our main results, where
we prove that there is an inverse critical temperature in a one-sided
long-range Ising model which is at most 8 times the critical inverse
temperature for the (two-sided) Ising model with long-range interactions.

Dynamical Systems