Let $R$ be a real closed field. We prove that if $R$ is uncountable, then any separately Nash (resp. arc-Nash) function defined over $R$ is semialgebraic (resp. 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. Joint work with W. Kucharz and A. El-Siblani.