  Thresholded hybrid systems are restricted dynamical systems, where the
current mode, and hence the ODE system describing its behavior, is solely
determined by externally supplied digital input signals and where the only
output signals are digital ones generated by comparing an internal state
variable to a threshold value. An attractive feature of such systems is easy
composition, which is facilitated by their purely digital interface. A
particularly promising application domain of thresholded hybrid systems is
digital integrated circuits: Modern digital circuit design considers them as a
composition of Millions and even Billions of elementary logic gates, like
inverters, GOR and Gand. Since every such logic gate is eventually implemented
as an electronic circuit, however, which exhibits a behavior that is governed
by some ODE system, thresholded hybrid systems are ideally suited for making
the transition from the analog to the digital world rigorous.
  In this paper, we prove that the mapping from digital input signals to
digital output signals is continuous for a large class of thresholded hybrid
systems. Moreover, we show that, under some mild conditions regarding
causality, this continuity also continues to hold for arbitrary compositions,
which in turn guarantees that the composition faithfully captures the analog
reality. By applying our generic results to some recently developed thresholded
hybrid gate models, both for single-input single-output gates like inverters
and for a two-input CMOS NOR gate, we show that they are continuous. Moreover,
we provide a novel thresholded hybrid model for the two-input NOR gate, which
is not only continuous but also, unlike the existing one, faithfully models all
multi-input switching effects.
