# Critical Values of multivariate polynomials are closed.

Consider a multivariate polynomial map $F:\mathbb{R}^n \rightarrow \mathbb{R}^n.$ Is it always true that the set $C$ in $\mathbb{R}^n$ of critical values is closed?

More specifically, the Sard's theorem tell us that measure of critical values is zero. But when dealing with Polynomials can one easily show that this set is closed nowhere dense?

In other words, does there exists an open dense set S of the real Euclidean space such that for all $c \in S,$ the polynomial system $F(x) -c =0$ has only non-singular roots?

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I have the same question about the set of critical values of $f$: $\mathbb{R^n}\rightarrow \mathbb{R}$ is finite. Someone help us, please! Thanks. –  user52523 Dec 14 '12 at 15:38
positive dimensional sets can exist –  user64883 Mar 4 at 3:34