The non-iterates are dense in the space of continuous self-maps

Abstract In this paper we first prove a nonexistence result on iterative roots, which presents several sufficient conditions for identifying self-maps arbitrary sets that have no roots of any order. Then, using result, when X is [ 0 , 1 stretchy="false">] m , $\mathbb{R}^m$?> R or $S^1$?> S every non-empty open set the space $\mathcal{C}(X)$?> continuous endowed with compact-open topology contains map does not even discontinuous order $n\geqslant 2$?> n ⩾ 2 . This, in particular, proves complement $\{f^n: f\in \mathcal{C}(X)~\text{and}~n\geqslant 2\}$?> non-iterates, dense these