Coanalytic sets and symmetric behavior in functions
A function \(f: R\to R\) (\(R\) the real line) is said to be smooth (symmetric, symmetrically continuous) at a point \(p\) if \(\lim_{h\to 0} (f(p+h)+ f(p-h)- 2f(p))/ 2h=0\) \((\lim_{h\to 0} (f(p+h)+ f(p-h)- 2f(p))=0\), \(\lim_{h\to 0} (f(p+h)- f(p-h))=0)\). It is said that \(f^ s(p)\) is the symmetric derivative of \(f\) at \(p\) if \(\lim_{h\to 0} (f(p+h)- f(p-h))/ 2h= f^ s(p)\). If \(f\) is a function, then \(SM(f)\), \(S(f)\), \(SD(f)\), and \(SC(f)\) denote the set of points where \(f\) is smooth, symmetric, symmetrically differentiable, and symmetrically continuous, respectively. The author proves: Let \(M\) be a zero- dimensional coanalytic subset of \(R\). Then there exists a Baire 2 function \(f\) such that \(M\) is homeomorphic to \(SM(f)\), \(S(f)\), \(SD(| f|)\), and \(SC(| f|)\). This statement improves in some sense one of author's earlier results [Proc. Am. Math. Soc. 119, No. 3, 915-923 (1993; Zbl 0791.26002)].
scientific article; zbMATH DE number 635941