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I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the corresponding vector field is continuously differentiable. I've also noticed that in most examples where unique solutions don't exist, the differentiability of the vector field is missing. My question is, will any weaker condition suffice to guarantee only the existence (and not necessarily the uniqueness) of an ODE? Is it for example true that if the vector field is just continuous, a solution exists even if not unique? I went through Existence of Solution to Differential Equations., but this doesn't seem to answer my doubts completely.

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Yes. Peano's existence theorem says that there exists (locally, as usual) at least one solution if the right-hand side is continuous.

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