Lectures related to Local-existence-unicity-theorem

Demonstration of Uniqueness Theorem

Presents a detailed proof of the uniqueness theorem for functions f and g.

Local Lipschitz Functions

Explores locally Lipschitz functions, discussing differentiability, unique solutions, and function reduction.

Gradient Descent: Lipschitz Continuity

Explores Lipschitz continuity in gradient descent optimization and its implications on function optimization.

Advanced Analysis 2: Continuity and Limits

Delves into advanced analysis topics, emphasizing continuity, limits, and uniform continuity.

Real Functions: Continuity Theorem

Covers the Continuity Theorem for functions dependent on a parameter, proving the continuity of a function g.

Intermediate value theorem

Explores uniform continuity, Lipschitz functions, and the intermediate value theorem with examples and proofs.

Initial Problem Solutions

Covers the description of problem solutions and the concept of compactness and uniform continuity.

Cauchy Problem: Initial Conditions

Discusses the Cauchy problem for ODEs with initial conditions and the importance of homeomorphism and Lipschitz continuity.

Differential Equations: Solutions and Periodicity

Explores dense sets, Cauchy sequences, periodic solutions, and unique solutions in differential equations.

Homogeneous Equations: Advanced Analysis II

Explores second-order linear scalar homogeneous equations in advanced analysis II.

Darboux Theorem: Advanced Analysis I

Explores the Darboux theorem for continuous functions on closed intervals, emphasizing uniform continuity and function behavior implications.

Uniform Continuity: Proof and Theorem

Covers the concept of uniform continuity and a theorem on continuous functions.

Advanced Analysis I: Continuous Functions on Compact Sets

Explores the necessity of uniform continuity for continuous functions on compact sets.

Existence of Solutions for Poisson-Dirichlet Problem

Covers the existence of solutions for the Poisson-Dirichlet problem, focusing on showing that certain conditions hold for locally bounded and Hölder continuous functions.

Advanced Analysis II: Hessians and Directional Derivatives

Explores Hessians, directional derivatives, and function continuity in advanced analysis.

Intermediate Value Theorem

Covers the Intermediate Value Theorem, uniform continuity, Lipschitz functions, and the properties of continuous functions.

Separation of Variables: Solving Differential Equations

Covers the method of separation of variables for solving differential equations, focusing on the construction and uniqueness of solutions.

Real Functions: Continuity Extension

Discusses extending a function uniformly and its continuity properties in real functions.

Limits and Continuity: Analysis 1

Explores limits, continuity, and uniform continuity in functions, including properties at specific points and closed intervals.

Cantor-Heine Theorem

Covers the Cantor-Heine theorem, discussing uniform continuity and compactness.