ABSTRACT. Several decompositions of continuity each stronger than Norman Levine's are found improving results of J. Chew and J. Tong, as well as of the first two named authors above.

We investigate the dense mapping property introduced by Keller in connection with iteration in Newton's method. Various kinds of functions are shown to have the dense mapping property. We show that a function has the dense mapping property iff it is bilaterally quasicontinuous. We also present an invariance theorem and other results on omega-limit sets.

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The properties of continuous stable iterated function systems are investigated, and equicontinuity is studied in relation to stable iterated function systems.