A metric space of discontinuous functions on the plane (Q756236)

The authors introduce a generalization of the Skorokhod space \(D[0,1]\) for the case when the domain of definition of the functions is the space \(X=[0,1]^ 2\). They define \(D(X)\) as the space of all functions \(f:X\to {\mathbb R}\) possessing the following property: for any point \(x\in X\) there exist two smooth curves \(\phi_ 1\) and \(\phi_ 2\) in \(X\) (depending on \(x\) and \(f\)) passing through \(x\) and non-tangential there, and such that \(\lim f(y)\) exists whenever y approaches \(x\) without crossing \(\phi_ 1\) and \(\phi_ 2\). A similar generalization has been introduced earlier by \textit{M. L. Straf} [Proc. 6th Berkeley Symp. Math. Statist. Probab., Univ. Calif. 1970, 2, 187--221 (1972; Zbl 0255.60019)] who required the existence of \(\lim f(y)\) when y approaches \(x\) without crossing the lines passing through \(x\) which are parallel to the coordinate axes. So the version of the space \(D(X)\) considered in the paper under review, as compared with the previous one, is wider and coordinate free. The authors equip \(D(X)\) with the appropriate metric \(d\) and show that the space \((D,d)\) is complete and separable. There are given examples of random fields with trajectories belonging to \(D(X)\).