Understanding Basic Automata Theory in the Continuous Time Setting

The development of paradigms, in which continuous time is involved in
cooperation with, or instead of, discrete time, is dominated by a
plethora of definitions, terminology and notation, which is not free
of ad-hoc and ambiguous decisions. Hence, the interst to isolate the
right concepts that capture the reality we want to understand.  From
this perspective we analyze some disputable concepts, keeping to the
following guidelines:

1. Building on basic automata theory.
2. Coherence with discrete-time paradigms whose continuous-time
analogs and/or mutants we would like to understand.
3. Functions, notably input/output behavior of devices, are more basic
than sets (languages) accepted by them.