Continuous function (set theory)

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages. 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,
$s_{\beta} = \limsup\{s_{\alpha}| \alpha < \beta\} = \inf \{ \sup\{s_{\alpha}| \delta \leq \alpha < \beta\} | \delta < \beta\} \,$
$s_{\beta} = \liminf\{s_{\alpha}| \alpha < \beta\} = \sup \{ \inf\{s_{\alpha}| \delta \leq \alpha < \beta\} | \delta < \beta\} \,.$