  This paper proposes a tractable family of remainder-form mixed-monotone
decomposition functions that are useful for over-approximating the image set of
nonlinear mappings in reachability and estimation problems. Our approach
applies to a new class of nonsmooth, discontinuous nonlinear systems that we
call either-sided locally Lipschitz semicontinuous (ELLS) systems, which we
show to be a strict superset of locally Lipschitz continuous (LLC) systems,
thus expanding the set of systems that are formally known to be mixed-monotone.
In addition, we derive lower and upper bounds for the over-approximation error
and show that the lower bound is achieved with our proposed approach, i.e., our
approach constructs the tightest, tractable remainder-form mixed-monotone
decomposition function. Moreover, we introduce a set inversion algorithm that
along with the proposed decomposition functions, can be used for constrained
reachability analysis and guaranteed state estimation for continuous- and
discrete-time systems with bounded noise.
