Differentiable Functions: Definitions and Properties

This lecture covers the definition and properties of differentiable functions, focusing on the concept of differentiability for a function f: I→R, where I is an open interval. The lecture explains the definition of differentiability, the existence of limits, and the relationship between differentiability and continuity. It also discusses partial derivatives and provides examples to illustrate the concepts. The importance of understanding differentiability in the context of rational functions is emphasized, along with methods for verifying differentiability. The lecture concludes with a discussion on the necessary conditions for differentiability and the uniform bound theorem.