  Entropy rate is a real valued functional on the space of discrete random
sources which lacks a closed formula even for subclasses of sources which have
intuitive parameterizations. A good way to overcome this problem is to examine
its analytic properties relative to some reasonable topology. A canonical
choice of a topology is that of the norm of total variation as it immediately
arises with the idea of a discrete random source as a probability measure on
sequence space. It is shown that entropy rate is Lipschitzian relative to this
topology, which, by well known facts, is close to differentiability. An
application of this theorem leads to a simple and elementary proof of the
existence of entropy rate of random sources with finite evolution dimension.
This class of sources encompasses arbitrary hidden Markov sources and quantum
random walks.
