Strong porosity features of typical continuous functions
A subset A of R is called bilaterally strongly porous at a point a, \(a\in R\), iff the right-porosity \(p_+(A,a)\) and the left-porosity \(p_- (A,a)\) of A at a are both 1. The right-porosity \(p_+(A,a)\) of A at a is the number: \(\limsup_{t\to 0+}\quad d(A,a,t)/t,\) where d(A,a,t) is the length of the largest open interval in \((x,x+t)-A.\) Their main result is the Theorem 2.5: For each \(\sigma\)-compact subset K of the space \(C(<0,1>)\) of all continuous functions on \(<0,1>\) with the supremum norm, the set \(F=\{f\in C(<0,1>):\) for each \(g\in K\) the set \(\{x\in <0,1>:\quad f(x)=g(x)\}\) is bilaterally strongly porous\(\}\) is residual in \(C(<0,1>)\). The authors give also three consequences of their result and concluding remarks. We give here their third consequence: For any system \(E=\{E_ x:\quad x\in <0,1>\}\) of paths, where \(E_ x\) is not bilaterally strongly porous at x for every \(x\in <0,1>\), the class of all functions of \(C(<0,1>)\) which are nowhere E-differentiable is an in \(C(<0,1>)\) residual set.
scientific article; zbMATH DE number 3950893