Some typical properties of symmetrically continuous functions, symmetric functions and continuous functions (Q1374551)

The existence of a bounded measurable (periodic) symmetrically continuous function and a bounded measurable symmetric function on an interval \([a,b]\) which are discontinuous at points of a set of power \(c\) was proved by D. Preiss and T.-C. Tran, respectively. It is used to show, in a straightforward way, that the functions with these properties are residual in the respective spaces of all bounded measurable symmetrically continuous functions (BSC[a,b]) and of all bounded measurable symmetric functions (BS[a,b]). A simple construction is made to obtain that the set of continuous functions for which \[ \lim \sup_{h\to 0} |[f(x+h)+f(x-h)-2f(x)]/h|=\infty \] for all \(x\in (0,1)\) is residual in \(C[0,1]\). This gives a weaker result than that which M.-J. Evans obtained for approximative \(\lim \sup\) and \(\lim \inf\). However, the construction of a particular function of Evans is not needed. It is remarked that, similarly, we can obtain an analogous weakening of the result of P. Kostyrko for \(\lim \sup_{h\to 0} |[f(x+h)-f(x-h)]/h|=\infty\) without using the construction of L. Filipczak.