  In the previous paper (Kiri Sakahara and Takashi Sato. Basic Topological
Concepts and a Construction of Real Numbers in Alternative Set Theory. arXiv
e-prints, arXiv:2005.04388, May 2020), the authors displayed basic topological
concepts and a construction of a system of real number in alternative set
theory (AST). The present paper is a continuation of that research providing
additional treatments of real functions. The basic properties of
differentiation in AST are preserved as in the conventional calculus.
