Let $p:\mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial with non-empty zero set $S$. Is it true that for any $x,y$ in the same connected component $C$ of $S$ there exists a piecewise smooth path $\gamma:[0,1] \rightarrow C$ such that $\gamma(0)=x$, $\gamma(1)=y$? The question arose in a calculus of variations problem with a specific $p:\mathbb{R}^5 \rightarrow \mathbb{R}$, but I suspect the answer is yes regardless of the specific polynomial.
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In their paper "Triangulations of semialgebraic sets", Ohmoto and Shiota improve the earlier result of Lojasiewicz (S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449-474.) and show that each real semialgebraic set admits a $C^1$-smooth triangulation, where smoothness is understood as smoothness of maps of closed simplices. This implies the statement you are asking since level sets of polynomials are algebraic.
