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Let $L_2=\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial.

If ${L_1} \subseteq {L_2}$ and $L_1$ is union of continuous curves. Can we say that $L_1$ is piecewise $C^∞$ curve?

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