Iterative square roots of functions

An iterative square root of a function $f$ is a function $g$ such that $g(g(\cdot ))=f(\cdot )$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. This is used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb{R}}^m$ and to the whole of ${\mathbb{R}}^m$ for every positive integer $m\mathrm{.}$ On the other hand, we also prove that every continuous self-map of a space homeomorphic to the unit cube in ${\mathbb{R}}^m$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.