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Yes, this is possible. Consider the intersection of the helicoid (the graph of $\tan^{-1}(y/x)$) with its tangent plane at $(1,0,0)$ (the graph of $y$) near $(1,0,0)$. The projection of the intersection curves to $\{z = 0\}$ consists of the line $y = 0$ and the curve $x = y/\tan(y) \sim 1 - y^2/3$, so near $(1,0,0)$ the projection to $\{z = 0\}$ of the union of $\gamma_1$ and $\gamma_2$ is roughly the $C^{1/2}$ graph $y = \sqrt{3(1-x)}$ for $x < 1$ and $y = 0$ for $x > 1$.