Critical sets in the plane
For a smooth \((C^ \infty)\) function \(f:\mathbb{R}^ m\to\mathbb{R}\) consider its critical set cp\((f)=\{x\in\mathbb{R}^ m:(Df)_ x=0\}\). A subset \(C\subset\mathbb{R}^ m\) is critical if \(C=\text{cp}(f)\) for some smooth \(f:\mathbb{R}^ m\to\mathbb{R}\), and it is properly critical if such an \(f\) exists which is proper. This very interesting paper deals with the following nice problem: which topological properties have a critical set, respectively a properly critical set? I mention here the main results: A compact nonempty subset of \(\mathbb{R}^ 2\) is critical if and only if it is properly critical if and only if the components of its complement are multiply connected (Theorem 1). The noncompact case is also studied: A closed set \(K\subset\mathbb{R}^ 2\) is critical if and only if \(\infty\) is arcwise accessible in each simply connected subset of \(K^ c\), the complement of \(K\) (Theorem 5). The authors mention some results concerning the above problem in general case \(m\geq 3\).
scientific article; zbMATH DE number 59885