Implicit Functions: Theory and Examples

This lecture covers the theory of implicit functions, focusing on the relationship between explicit and implicit definitions. Starting with the definition of implicit functions, the instructor explains the concept of separability and differentiability. The lecture delves into the neighborhood of a function and the classification of functions based on their properties. The theorem of implicit functions is illustrated through various examples, emphasizing the conditions for the existence and differentiation of implicit functions. The lecture concludes with a detailed explanation of the derivative of implicit functions and the proof of theorems related to implicit functions.