On the sectionwise connectedness of a contingent (Q662653)

The paper deals with the study of connectedness of a contingent (tangent cone) of a continuous mapping \(f : {\mathbb R} \rightarrow X\), where \(X\) is a real normed space. It is well-known that the contingent of each nonempty subset of a vector space, in particular the contingent \(T_f(0)\subset {\mathbb R}\times X\) of the graph \(G(f)\) at the point \((0,f(0))\), is connected. The aim of the paper is to investigate the connectedness of the sections of \(T_f(0)\) by hypersurfaces which do not pass through \((0, 0_X)\). It is shown that such sections need not to be connected. To prove it, the right unit hemisphere \(S^{+}\) is taken as a hypersurface. The cases \(\dim\, X < \infty\) and \(\dim\, X = \infty\) are discussed separately.