Functions whose set of critical points is an arc (Q2435093)

Let \(M\) be a \(C^\infty\) closed connected manifold of dimension \(\dim(M) \geq 2\) and let \(\mathcal{J}\) consist of all functions \(f \in C^1(M,\mathbb{R})\) such that the set \(\mathrm{Crit}(f)=\{x \in M: \text{ all directional derivatives of }f\text{ at }x\text{ are }0\}\) is an arc (i.e., homeomorphic to the unit interval \(I=[0,1]\)) and the restriction \(f|_{\mathrm{Crit}(f)}\) is nowhere locally constant on \(\mathrm{Crit}(f)\). It is shown that \(\mathcal{J}\) is dense in \(C^0(M,\mathbb{R})\). The proof is based on a construction due to [\textit{T. W. Körner}, J. Lond. Math. Soc., II. Ser. 38, No. 3, 442--452 (1988; Zbl 0726.26008)]. The result is applied to flows on manifolds.