Optimal version of the Picard-Lindel\"of theorem

Consider the differential equation y = F (x, y). We determine the weakest possible upper bound on |F (x, y)−F (x, z)| which guarantees that this equation has for all initial values a unique solution, which exists globally. Let F : R → R be a continuous function. The well known global Picard-Lindelöf theorem states that if F is Lipschitz continuous with respect to the second variable, then for every real number y0, the initial value problem y ′ = F (x, y), y(0) = y0 has a unique solution, which exists globally. On the other hand the initial value problem y′ = 2 √ |y|, y(0) = 0 has infinitely many solutions, which can be parametrized by real numbers −∞ ≤ a ≤ b ≤ ∞ as