A discontinuous group is a mathematical concept relating to mappings in topological space.
Definition Let T {\displaystyle T} be a topological space of points τ {\displaystyle \tau } , and let τ → f ( τ , x ) {\displaystyle \tau \to f(\tau ,x)} , x ∈ G {\displaystyle x\in G} , be an open continuous representation of the topological group G {\displaystyle G} as a transitive group of homeomorphic mappings of T {\displaystyle T} on itself. The representation τ → f ( τ , a ) {\displaystyle \tau \to f(\tau ,a)} a ∈ H {\displaystyle a\in H} of the discrete subgroup H ⊂ G {\displaystyle H\subset G} in T {\displaystyle T} is called discontinuous, if no sequence f ( τ , a n ) {\displaystyle f(\tau ,a_{n})} ( n = 1 , 2 , … {\displaystyle n=1,2,\ldots } ) converges to a point in T {\displaystyle T} , as a n {\displaystyle a_{n}} runs over distinct elements of H {\displaystyle H} .
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