Functions on products $X \times Y$ with applications to Ascoli spaces, $k_{\mathbb{R}}$-spaces and $s_{\mathbb{R}}$-spaces

Abstract: We prove that a Tychonoff space $X$ is (sequentially) Ascoli iff for every compact space $K$ (resp., for a convergent sequence $\mathbf{s}$), each separately continuous $k$-continuous function $\Phi:X\times K\to \mathbb{R}$ is continuous. We apply these characterizations to show that an open subspace of a (sequentially) Ascoli space is (sequentially) Ascoli, and that the $\mu$-completion and the Dieudonn\'{e} completion of a (sequentially) Ascoli space are (sequentially) Ascoli. We give also cover-type characterizations of Ascoli spaces and suggest an easy method of construction of pseudocompact Ascoli spaces which are not $k_\mathbb{R}$-spaces and show that each space $X$ can be closely embedded into such a space. Using a different method we prove Hu\v{s}ek's theorem: a Tychonoff space $Y$ is a locally pseudocompact $k_\mathbb{R}$-space iff $X\times Y$ is a $k_\mathbb{R}$-space for each $k_\mathbb{R}$-space $X$. It is proved that $X$ is an $s_\mathbb{R}$-space iff for every locally compact sequential space $K$, each $s$-continuous function $f:X\times K\to\mathbb{R}$ is continuous.