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I'm wondering if the continuity of $f(t,y)$ is necessary for the existence and uniqueness of the solution to $dy/dt=f(t,y(t))$.

I think the existence and uniqueness only require $f(t,y)$ has a continuous second partial differential.

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    You should say what you have edited in your question such that the answer posted do not appear as being wrong. And assuming continuous second partial differentiation is a stronger condition than Lipschitz continuity.2011-06-10

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