This technical note considers input-to-state stability analysis of discrete-time systems using continuous Lyapunov functions. The main result reveals a relation between existence of a continuous Lyapunov function and inherent input-to-state stability on compact sets with respect to both inner and outer perturbations. If the Lyapunov function is K8-continuous, the result applies to unbounded sets as well.
Lazar, M., Heemels, W. P. M. H., & Teel, A. R. (2013). Further input-to-state stability subtleties for discrete-time systems. IEEE Transactions on Automatic Control, 58(6), 1609-1613. https://doi.org/10.1109/TAC.2012.2231611