Continuous function (set theory)

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In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the union or supremum of all ordinals in previous ones. More formally, let γ be an ordinal, and $s := \langle s_{\alpha}| \alpha < \gamma\rangle$ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ $, s_{\beta} = \sup\{s_{\alpha}| \alpha < \beta\}.$ Alternatively, s is continuous if s: γ → range(s) is a continuous function when the sets are each equipped with the order topology.