An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : –1,1] with (0) = x and f increasing on –1, 1]. Thenfor every there exists a continuous map : 0, 1) suchthat (0) = y, f is increasing on 0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.