The Pontryagin maximum principle. Ab ovo usque ad mala

A proof of the Pontryagin maximum principle for a sufficiently general optimal control problem is presented; the proof is based on the implicit function theorem and the theorem on the solvability of a finite-dimensional system of nonlinear equations. The exposition is self-contained: all necessary preliminary facts are proved. These facts are mainly related to the properties of solutions to differential equations with discontinuous right-hand side and are derived as corollaries to the implicit function theorem, which, in turn, is a direct consequence of Newton's method for solving nonlinear equations.