The composition of two connected \(G_\delta\) functions has a fixed point (Q1769726)

It is well-known that if \(f,g\colon [0,1]\to [0,1]\) are continuous then their composition has a fixed-point. \textit{M.~Csörnyei, T.C.~O'Neil} and \textit{D.~Preiss} [Real Anal. Exch. 26, No.~2, 749--760 (2001; Zbl 1011.26004)] and independently \textit{M.~Elekes, T.~Keleti}, and \textit{V.~Prokaj} [Real Anal. Exch. 27, No. 1, 131--140 (2002; Zbl 1011.26006)] proved that the same result holds for \(DB_1\) functions. The author of the paper under review shows a more general result, namely the analogous result holds also for pairs of functions with connected \(G_{\delta}\) graphs. Remark. Recently the same author has shown that the composition of \(n\) connected \(G_{\delta}\) functions, each of which maps the unit interval into itself, must have a fixed-point [see \textit{P.~Szuca}, Real Anal. Exch. 29(2003/04), No.~1, 205--209 (2004; Zbl 1065.26004)].