Title: Regularity of Lie Groups,
Abstract: We solve the regularity problem for Milnor's infinite dimensional Lie groups
in the $C^0$-topological context, and provide necessary and sufficient
regularity conditions for the standard setting ($C^k$-topology). We prove that
the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$
the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the
evolution map is defined on all smooth curves, then $G$ is Mackey complete -
This is a completeness condition formulated in terms of the Lie group
operations that generalizes Mackey completeness as defined for locally convex
vector spaces. Then, under the presumption that $G$ is locally $\mu$-convex, we
show that each $C^k$-curve, for $k\in \mathbb{N}_{\geq
1}\sqcup\{\mathrm{lip},\infty\}$, is integrable (contained in the domain of the
evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and
$\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the
Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special
type of sequence consisting of piecewise integrable curves - A similar result
is proven for the case $k\equiv 0$; and we provide several mild conditions that
ensure that $G$ is $\mathrm{k}$-confined for each $k\in
\mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the
differentiation of parameter-dependent integrals in the standard topological
context. In particular, we show that if the evolution map is well defined and
continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$,
then it is smooth thereon $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is
$\hspace{0.2pt}$ Mackey complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}$
$\hspace{1pt}/\hspace{1pt}$ integral complete for $k\equiv 0$. This result is
obtained by calculating the directional derivatives explicitly - recovering the
standard formulas that hold in the Banach case.