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* <math>f</math> is '''continuous''': for every limit ordinal <math>\lambda</math> (i.e. <math>\lambda</math> is neither zero nor a successor), <math>f(\lambda)=\sup\{f(\alpha):\alpha<\kappa\}</math>.
It can be shown that if <math>f</math> is normal then <math>f</math> commutes with [[supremum|suprema]]; for any nonempty set <math>A</math> of ordinals,
:<math>f(\sup A)=\sup f(A) = \sup\{f(\alpha):\alpha
Indeed, if <math>\sup A</math> is a successor ordinal then <math>\sup A</math> is an element of <math>A</math> and the equality follows from the increasing property of <math>f</math>. If <math>\sup A</math> is a limit ordinal then the equality follows from the continuous property of <math>f</math>.
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