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I would like to prove the following statement: Assume [itex]f\in C^{1}(\mathbb{R})[/itex]. Then the initial value problem [itex]\dot{x} = f(x),\quad x(0) = x_{0}[/itex] has a unique solution, on any interval on which a solution may be defined.

I haven't been able to come up with a proof myself, but would really like to see a direct proof, not using too serious tools from a sophisticated theory of ODE's. I would very much appreciate it if someone could help me out.

Thanks in advance!

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# How to prove: Uniqueness of solution to first order autonomous ODE

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