Separately Nash and arc‐Nash functions over real closed fields

Let R be a real closed field. We prove that if R is uncountable, then any separately Nash (respectively, arc‐Nash) function defined over R is semialgebraic (respectively, continuous semialgebraic). To complete the picture, we provide an example showing that the assumption on R to be uncountable cannot be dropped. Moreover, even if R is uncountable but non‐Archimedean, then the shape of the domain of a separately Nash function matters for the conclusion. For R = R , we prove that arc‐Nash functions coincide with arc‐analytic semialgebraic functions.