The questions are about "local non-satiation"

Q1. Given a real-valued function F(x) defined on a CLOSED set D, a non-empty subset of X, how can I verify that this is a function representing a preference order that is locally non-satiated on D?

Q2. Which work in the references of the book is of the problem of local non-satiation? Is it (Arrow and Hahn, 1971)? But Takayama said in Mathematical Economics (2nd Ed. 1985) that Koopmans had first used it in a work of 1957.

A suggestion:

Make a reference note for each chapter or show it clearly at the end of each reference so that readers can easily find the right reference for something they have further interests.