An illustrative example of complete connected space with a continuous function having an extremum at every point

摘要

We construct a complete metric space "realized" in R-3 at which the height function has a local (nonstrict) extremum at every point. The question of the existence of such a space was posed in [1]. Such examples were constructed (see, for example, [3,2]) but seem to be less transparent since they are "realized" in R-n = 4, and are hard to draw. (C) 2020 Published by Elsevier B.V.