Symmetric and smooth functions: A few questions and fewer answers (Q1062153)

Let f be a real function of one real variable. We say that f is symmetric or smooth at x if \(\Delta^ 2f(x,h)=f(x+h)+f(x-h)-2f(x)=o(1)\) respectively \(\Delta^ 2f(x,h)=o(h)\) as \(h\to 0\). The notions of approximately symmetric, approximately smooth, \(L_ p\)-symmetric and \(L_ p\)-smooth functions have been defined by Neugebauer in a natural manner. The author summarizes the results concerning the continuity and differentiability properties of functions which are symmetric or smooth at every point in the above-mentioned sense. He draws up six open problems, e.g. must the set of points of discontinuity of a symmetric function be \(\sigma\)-porous?