Derivability and Continuity

This lecture covers the concepts of derivability and continuity, exploring functions that are continuously differentiable but not twice differentiable, using examples to illustrate the properties of functions at different points. The instructor discusses the local nature of continuity and derivability, emphasizing the importance of understanding the limits and derivatives of functions to determine their properties. The lecture also delves into the notion of lateral derivatives and provides geometric interpretations of derivatives, showcasing functions that exhibit specific behaviors at different points. Additionally, the lecture touches upon the hierarchy of differentiability levels and the implications of functions being continuously differentiable. Through various examples and explanations, the instructor demonstrates how to analyze functions in terms of their derivability and continuity.