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Realizing a set as the image of a smooth map

Consider the following subset of $\mathbb{R}^2$: $S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$ (See here for a plot on Wolfram Alpha.) The origin $(0, 0)$ is a kind of singular point of $S$ …
Nicolas Boumal's user avatar