  It is well-known that there always exists at least one stationary measure
compatible with a continuous g-function g. Here we prove that if the set of
discontinuities of the g-function g has null measure under a candidate measure
obtained by some asymptotic procedure, then this candidate measure is
compatible with g. We explore several implications of this result, and discuss
comparisons with the literature concerning assumptions and examples. Important
part of the paper is concerned with the case of variable length memory chains,
for which we obtain existence, uniqueness and weak-Bernoullicity (or
$\beta$-mixing) under new assumptions. These results are specially designed for
variable length memory models, and do not require vanishing uniform variation.
We also provide a further discussion on some related notions, such as random
context processes, non-essential discontinuities, and finally an example of
everywhere discontinuous stationary measure.
