Discrete subgroups and Discontinuous subgroups of the isometries of the euclidean plane I am working with subgroups of $\operatorname{Iso}(\mathbb{R}^2)$ (isometries on $\mathbb{R}^2$), and am trying to classify Frieze and Wallpaper groups. I have two books, which use a different word to (from what I can see) mean the same thing:
$G \leqslant \operatorname{Iso}(\mathbb{R}^2)$ is $\textbf{discrete}$ if it does not contain rotations of arbitrarily small angle an no translations of arbitrarily small distance.
$G \leqslant \operatorname{Iso}(\mathbb{R}^2)$ is $\textbf{discontinuous}$ if for every $x \in \mathbb{R}^2$ there is a neighbourhood $U$ of $x$ such that $Orb_{G}(x) \cap U = \{x\} $
Firstly, are these definitions standard? (Are they consistent with more general definitions?). Are they equivalent (I can see how discontinuous $\Rightarrow$ discrete but I don't know about the other way)
 A: The definitions are standard. Discrete groups are in general defined as follows:
Definition: A discrete subgroup of a topological group $G$ is a subgroup $H$ such that there is an open cover of $H$ in which every open subset contains exactly one element of $H$; in other words, the subspace topology of $H$ in $G$ is the discrete topology.  
For subgroups of isometries also the following definition is used:
A subgroup $\Gamma\in \operatorname{Iso}(\mathbb{R}^n)$ is called discrete if
each orbit $\Gamma x$ for the action of $\Gamma$ on $X=\mathbb{R}^n$ is a discrete subset of $\mathbb{R}^n$.
The definition is equivalent, for $n=2$, to the given one.
Definition: An action of a discrete group $G$ on a topological space $X$ is properly discontinuous if and only if $\forall x,y \in X$ there exist neighbourhoods $U_x$ and $U_y$ of $x$ and $y$ respectively such that the set $\{g \in G : g \cdot U_x \cap U_y\ \neq \emptyset\}$ is finite.
In general, "properly discontinuous" implies "discrete", but the converse is not true in general. It is true, however, for actions of $\operatorname{Iso}(\mathbb{R}^n)$. See also the question 
Discrete subgroups of isometry group $\mathbb{R}^n$.
