# Connectedness properties of real algebraic set

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.

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.