On locally symmetric and locally Jensen functions (Q1189103)

A function \(f:R\to R\) is said to be locally symmetric at a point \(x\) if there exists \(\delta=\delta(x)>0\) such that (S) \(f(x+h)=f(x-h)\) holds for every \(h\), \(0<h<\delta\); \(f\) is said to be locally symmetric, if it is locally symmetric at each \(x\in R\). The function \(f:R\to R\) is said to be uniformly locally symmetric on a set \(A\), if there exists \(\delta>0\) such that (S) holds for each \(h\), \(0<h<\delta\), and for each \(x\in A\); \(f\) is said to be locally Jensen at \(x\in R\) if there exists \(\delta=\delta(x)>0\) such that (J) \((1/2)(f(x+h)+f(x-h))=f(x)\); \(f\) is said to be uniformly locally Jensen on the set \(A\subset R\) if there exists \(\delta>0\) such that (J) holds for each \(h\), \(0<h<\delta\), and \(x\in A\). Answering a question raised by the reviewer [Comput. Math. Appl. 17, 103- 115 (1989; Zbl 0703.26003)] and another similar one concerning locally Jensen functions, the following theorems are proved. Theorem 1. Let \(f\) be locally symmetric. Then \(f\) is a constant function iff there is a set \(A\) dense in \(R\) such that \(f\) is uniformly locally symmetric on \(A\). Theorem 2. If \(f\) is locally Jensen then \(f\) is a Jensen function iff there is a set \(A\) dense in \(R\) such that \(f\) is uniformly locally Jensen on \(A\).