Symmetric porosity, dimension, and derivates
If \(A\subset R\) \((R\) -- the real line) and \(x\in R\), then the symmetric porosity of \(A\) at \(x\) is defined as \(\limsup_{r\to 0+}\lambda(A,x,r)/r\), where \(\lambda(A,x,r)\) is the supremum of all positive numbers \(h\) such that there is a positive number \(t\) with \(t+h\leq r\) such that both of the intervals \((x-t-h,x-t)\) and \((x+t,x+t+h)\) lie in the complement of \(A\). A set \(A\) is symmetrically porous if it has positive symmetric porosity at each of its points and is called strongly symmetrically porous if it has symmetric porosity one at each of its points. The author constructs a strongly symmetrically porous closed set \(E\subset[0,1)\) which has Hausdorff dimension one. It is shown that there is a continuous monotone function \(f:[0,1]\to R\) such that \(f\) is differentiable at each \(x\in[0,1)-E\) and \(f^ -(x)\neq f^ +(x)\) at each \(x\in E-\{0\}\) \((f^ -(x)\) \([f^ +(x)]\) --- the upper left [right] derivate of \(f\) at \(x)\).
strongly symmetrically porous closed set
scientific article; zbMATH DE number 54510