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Consider the IVP $\mathbf{\dot x} = f(\mathbf{x},t)$ where $\mathbf{x}(0) = 0$, $f$ is continuous in some neighborhood of $(x,t) = (0,0)$ and $|f(x,t)-f(y,t)| \leq \frac{|x-y|}{t^\alpha}$. I would like to show the uniquenesss of the solution if $\alpha\in (0,1)$.

It seems like I would like to show that $f$ is locally Lipschitz in $x$ when $t = 0$ if $\alpha\in(0,1)$, (since it is clearly locally Lipschitz for all $t > 0$) but I haven't been able to tackle that by elementary methods (or any other, really). Another idea might be to bound the sequence of Picard iterates, but that seems very messy. A third idea would be to appeal to general facts about moduli of continuity. The fact that a solution is unique when the integral $\int_0^x\frac{dx}{\omega(x)}$ diverges for the modulus $\omega$ seems almost useful, except that in this case my modulus is a function of both $x$ and $t$. Am I in the right ballpark with these approaches, or is there something very obvious that I'm missing?

Also, this is homework, so I would be grateful for a small hint rather than a full solution. Thanks in advance!

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