Methodical Analysis of the Implicit Function Theorem

Abstract

This paper provides a systematic analysis of the Implicit Function Theorem, intended as a case study for learning mathematical analysis. The discussion covers classical formulations, key assumptions, generalizations, and applications, with an emphasis on opportunities to apply active learning methods at each stage. This analysis is particularly inspired by the works of Sierpińska (1994), Krantz and Parks (2013), and Freeman et al. (2014). As intended by the author, a scenario is presented illustrating how the concept of the implicit function can be effectively introduced to students.