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Abstract

In this paper, we study initial value problems for a class of functional equations. We introduce the concept of appropriate initial sets to enable the unique extension of an initial function into a solution defined over larger domains. Our analysis characterizes the structure and size of these initial sets, illustrating how functions specified on them can be uniquely extended to broader regions. Furthermore, we propose the Principle of Exclusion of Neighborhoods of Limit Points (PENLP)-a novel conceptual framework that elucidates the behavior of solutions near limit points and their impact on the extension process. We demonstrate that the initial sets must avoid the limit points of the functional equations to ensure uniqueness and well-posedness. The functional equations examined in this study include: $$ y(x+1)= y(bx),\quad b \neq 0, $$ $$ y(x)=y(2x),\quad y(x)=y(-2x),\quad y(3x)=y(x)+y(2x), $$ as well as the equations characterizing even functions and odd functions.