In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory: Carathéodory's theorem (conformal mapping), about the extension of conformal mappings to the boundary Carathéodory's theorem (convex hull), about the convex hulls of sets in Euclidean space Carathéodory's existence theorem, about the existence of solutions to ordinary differential equations Carathéodory's extension theorem, about the extension of a measure Borel–Carathéodory theorem, about the boundedness of a complex analytic function Carathéodory–Jacobi–Lie theorem, a generalization of Darboux...

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory. == Introduction == Consider the differential equation y ′ ( t )...