Convex Functions: Analytical Definition and Geometric Interpretation

This lecture covers the properties of functions such as monotonicity, parity, periodicity, extrema, and convexity. The instructor explains the concept of convexity both geometrically and analytically, illustrating how to define a convex function using a parameter lambda. Through examples involving absolute value and exponential functions, the lecture demonstrates how to determine convexity analytically. The concept of limits and lateral limits is introduced, with examples showing how to sandwich complex functions between simpler 'gendarme' functions to evaluate limits. The lecture concludes with a discussion on functions that may have limits from one side but not the other, emphasizing the importance of understanding limits for various types of functions.