We investigate the continuity of the ω-functions and real functions defined by weighted finite automata (WFA). We concentrate on the case of average preserving WFA. We show that every continuous ω-function definable by some WFA can be defined by an average preserving WFA and then characterize minimal average preserving WFA whose ω-function or ω-function and real function are continuous. We obtain several algorithmic reductions for WFA-related decision problems. In particular, we show that deciding whether the ω-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.

41 pages, 9 figures