The closure of the set of iterable functions (Q933713)

A function \(f:I\to I\) is iterable if it is embeddable into a continuous iteration semigroup \(F\) such that \(F(1,x)=f(x) \). Recall that the continuous iteration semigroup is a function \(F:(0,\infty )\times I\to I\) that is continuous with respect to each variable and such that \(F(t,F(s,x))=F(s+t,x)\). Characterization of iterable selfmappings of a real compact interval was given in [\textit{M. C. Zdun}, Pr. Nauk. Uniw. Sl. Katow. 308, (1979)]. The present paper describes the closure of the set of all iterable functions in the space of all continuous selfmappings of \(I\), where \(I\) is a real compact interval, endowed with the metric of uniform convergence. Finally, the author gives a comparison of various notions of near-embeddability.