On Carathéodory’s conditions for the initial value problem
Daniel C. Biles, Paul Binding · Proceedings of the American Mathematical Society · 1997
We prove a local existence theorem of Carathéodory–Goodman type for x ˙ ( t ) = f ( t , x ( t ) ) \dot {x}(t)=f(t,x(t)) where instead of f ( t , α ) f(t,\alpha ) being continuous in α \alpha we require only that it have no “downward discontinuities.”