A converse to a theorem of Sierpinski on almost symmetric sets (Q923741)

The author gives a general view on the symmetry of real functions. A collection of sets of reals, \({\mathcal L}\), is called a ``notion of largeness'' if it is closed under supersets and a set is called large if it is in \({\mathcal L}\). A real function f is said to be \({\mathcal L}\)-symmetric (locally \({\mathcal L}\)-symmetric) if for each real x, the set of h for which \(f(x+h)=f(x-h)\) is large (large in some neighborhood of zero). The \({\mathcal L}\)-symmetric derivative of f at x is defined to be \(\lim_{h\to 0}^{{\mathcal L}}(f(x+h)-f(x-h))/2h,\) where \(\lim_{h\to 0}^{{\mathcal L}}\) denotes that a set of h's ignored in the limit is the complement of a large set. A suitable choice of \({\mathcal L}\) gives some known kinds of the symmetry of real functions (full symmetry, almost symmetry, approximate symmetry, etc.). It is proved, under some conditions on \({\mathcal L}\), that monotonicity theorems for the \({\mathcal L}\)-symmetric derivative hold.