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If we have a complete partial order (i.e. directed complete) I find frequently the following definition of a continuous function. A function $f:A\to B$ where $A$ and $B$ are cpos is called continuous if it maps the suprema of directed subsets of $A$ (if exist) to the corresponding suprema of directed subsets of $B$.

In complete lattices I would define continuous functions as functions which preserve suprema and infima (since both exist in a complete lattice for any subset).

Since cpos are complete lattices are cpos the following question arises: Are both definitions consistent? The requirement that all suprema and infima are preserved is stronger than the requirement that only suprema of directed sets are preserved. Therefore it might be possible that both definitions are different. Or are they equivalent?

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# Definition of continuous functions in order theory

If we have a complete partial order (i.e. directed complete) I find frequently the following definition of a continuous function. A function $f:A\to B$ where $A$ and $B$ are cpos is called continuous if it maps the suprema of directed subsets of $A$ (if exist) to the corresponding suprema of directed subsets of $B$.

In complete lattices I would define continuous functions as functions which preserve suprema and infima (since both exist in a complete lattice for any subset).

Since cpos are complete lattices the following question arises: Are both definitions consistent? The requirement that all suprema and infima are preserved is stronger than the requirement that only suprema of directed sets are preserved. Therefore it might be possible that both definitions are different. Or are they equivalent?