One of the first big theorems you learn in ODEs is the following existence and uniqueness theorem:

Let $I$ be a closed rectangle in $\mathbb{R} \times \mathbb{R}$, and let $f: I \rightarrow \mathbb{R}$ be differentiable on the interior, and also assume that $\frac{\partial f}{\partial y}$ is continuous on $I$. Then for any point $(x_0,y_0)$ in the interior of $I$, there is a small interval $(x_0 - t, x_0 + t)$ about $x_0$ on which there exists a unique differentiable function $g: (x_0 - t, x_0 + t) \rightarrow \mathbb{R}$ such that $g'(x) = f(x,g(x))$ and $g(x_0) = y_0$.

My question is, can any sort of converse to this be stated? If not, are there any easy examples where you have a unique function $g$ satisfying the given properties, but without everything in the hypothesis holding?


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