  We study multifractal decompositions based on Birkhoff averages for sequences
of functions belonging to certain classes of symbolically continuous functions.
We do this for an expanding interval map with countably many branches, which we
assume can be coded by a topologically mixing countable Markov shift. This
generalises previous work on expanding maps with finitely many branches, and
expanding maps with countably many branches where the coding is assumed to be
the full shift. When the infimum of the derivative on each branch approaches
infinity in the limit, we can directly generalise the results of the full
countable shift case. However, when this does not hold, we show that there can
be different behaviour, in particular in cases where the coding has finite
topological entropy.
