On continuity of symmetric restrictions of Borel functions

Abstract:

We prove that if $X$ is a complete metric space dense-in-itself, $Y$ is a compact metric space and $F:X \times X\backslash \left \{ {(x,x):x \in X} \right \} \to Y$ is a Borel-measurable function such that $F({x_1},{x_2}) = F({x_2},{x_1})$ for every ${x_1},{x_2} \in X,{x_1} \ne {x_2}$, then there is a perfect subset $P$ of $X$ such that $F$ is uniformly continuous on $P \times P\backslash \left \{ {(x,x):x \in P} \right \}$. An immediate corollary of the above result is the following theorem proved by Bruckner, Ceder and Weiss: If $F$ is a real continuous function defined on a perfect set $P \subset R$, there is a perfect subset $Q$ of $P$ such that $f|Q$ has a derivative (finite or infinite) at every point of $Q$.