  We investigate the continuity of the \omega-functions and real functions
defined by weighted finite automata (WFA). We concentrate on the case of
average preserving WFA. We show that every continuous \omega-function definable
by some WFA can be defined by an average preserving WFA and then characterize
minimal average preserving WFA whose \omega-function or \omega-function and
real function are continuous.
  We obtain several algorithmic reductions for WFA-related decision problems.
In particular, we show that deciding whether the \omega-function and real
function of an average preserving WFA are both continuous is computationally
equivalent to deciding stability of a set of matrices.
  We also present a method for constructing WFA that compute continuous real
functions.
