Hybrid automata have been proposed as a language for modelling and analysing the interaction of digital and analogue dynamics in embedded computer systems. In the paper, hybrid automata are studied from a dynamical systems perspective. Extending earlier work on conditions for existence and uniqueness of executions of hybrid automata, we characterise a class of hybrid automata whose executions depend continuously on the initial state. The continuity conditions are subsequently used to derive an extension of LaSalle's principle for studying the stability of invariant sets of states of hybrid automata.