Local Inversion Theorem

This lecture covers the local inversion theorem, which states that for a function of class C², if the determinant of its derivative is non-zero, then it is a local diffeomorphism. The proof involves showing the existence of local inverses and the bijection and continuity of the inverse. Applications of the fixed-point theorem are also discussed, demonstrating the uniqueness of solutions. The lecture concludes with examples illustrating the continuity and uniform continuity of functions. Various matrices and vectors are used to explain the concepts.