The tangent approach to the characterization of the set of singular boundary points for an arbitrary function (Q1760956)

Let \(\mathbb{R}_+^3\) be the upper half-space of \(\mathbb{R}^3\), \(P\) a compact Hausdorff space and \(f:\mathbb{R}_+^3\to P\) a function. The author defines the set of singular boundary points of \(f\) as \[ E(f)=\bigcup_{a,b,c\in\mathbb{Q}} \{ \zeta \in\partial\mathbb{R}_+^3 \mid C(f,\zeta ,U(\zeta,c))\setminus C(f,\zeta ,V(\zeta ,a,b))\neq\emptyset\} , \] where \(C(f,\zeta ,U)\) is the cluster set of \(f\) at a point \(\zeta \) with respect to a set \(U\), \(U(\zeta,b)\) is the ellipsoid \((x_1-\zeta _1)^2+(x_2-\zeta _2)^2+b^2(x_3-1)^2\leq b^2\), and \(V(\zeta ,a,b)=U(\zeta ,b)\setminus\text{int} \, U(\zeta ,a)\). The main result of this note says that the set \(E(f)\) is perfect \(\sigma \)-porous.