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For a first order Diff Equa. x'=f(x,t) and the IC: x(0)=x_0.

with t from [0 to infinity)

If f(x,t) doesn't satisfy the Lipschitz condition, can I say for sure that there doesn't exist a global unique solution?

I think the answer is "no" but I am not sure. Can you all confirm?

Also, can I use the Lipschitz condition to check the existence of a local solution around the IC? I see somebody often check the continuity of f(x) and df(x)/dx around the IC. Is this equavilent to the Lipschitz?

Thanks

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# Existence of a local solution

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