Critical points of arbitrary mappings of \(\mathbb{R}^n\) into \(\mathbb{R}^m\)
This article develops some ideas and results due to \textit{M. de Guzman} [``Differentiation of integrals in \(\mathbb{R}^n\)'' (1975; Zbl 0327.26010)]. Given \(f: \mathbb{R}^n\to \mathbb{R}^m\), \(x\in\mathbb{R}^n\) is a critical point for \(f\) if \(\lim_{r\to 0}((\text{mes }f(V_r(x)))/ (\text{mes }V_r(x))^{m/n})= 0\), where mes is the Lebesgue exterior measure and \(V_r(x)\) is a ball of radius \(r\) centered at \(x\). Replacing lim by \(\liminf\), we get the notion of a weakly critical point for \(f\). Denoting by \(g(f)\) (respectively \(G(f)\)) the set of critical (weakly critical, respectively) point of \(f\), we have, for \(f\in C^1(\mathbb{R}^n; \mathbb{R}^n)\), \(g(f)= G(f)=\) the set of critical points (in the ordinary sense) of \(f\). Theorem 1 gives a characterization of \(G(f)\) and of \(g(f)\) for \(f\) arbitrary, in terms of some regularly contracting systems of subsets. A notion of order of a critical (weakly critical) point is introduced, with respect to which Theorem 2 gives a characterization of points which are critical (weakly critical) of a given order. Denoting by \(g_\alpha(f)\) the set of all weakly critical points \(f\), whose order exceeds \(\alpha\), Theorem 3 asserts that \(\text{mes }f(g_r(f))= 0\), where \(r= \max(0;n- m)\). A generalization to metric spaces is also proposed.
scientific article; zbMATH DE number 1188099