Continuous function (set theory)

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In

) of all values at previous stages. More formally, let γ be an ordinal, and ${\displaystyle s:=\langle s_{\alpha }|\alpha <\gamma \rangle }$ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ,

${\displaystyle s_{\beta }=\limsup\{s_{\alpha }:\alpha <\beta \}=\inf\{\sup\{s_{\alpha }:\delta \leq \alpha <\beta \}:\delta <\beta \}}$

and

${\displaystyle s_{\beta }=\liminf\{s_{\alpha }:\alpha <\beta \}=\sup\{\inf\{s_{\alpha }:\delta \leq \alpha <\beta \}:\delta <\beta \}\,.}$

Alternatively, if s is an

.

A normal function is a function that is both continuous and strictly increasing.