= Continuous group cohomology
{title2=$H_c^n(G,A)$}
In the continuous-cochain convention, <continuous group cohomology> of a <topological group> $G$ with coefficients in a topological <abelian group> $A$ uses continuous maps $G^n\to A$ and the usual <group cohomology> differential. The action of $G$ on $A$ must be continuous. In degree one, a continuous cocycle satisfies
$$
c(gh)=c(g)+g\cdot c(h),
$$
and one takes its class modulo maps $c(g)=g\cdot a-a$. Requiring continuity is substantive additional data, including the topology chosen on $A$.<Some complements to the Lazard isomorphism by Annette Huber and others>[Introduction and section 2.7, continuous cohomology in the p-adic setting]{ref}
For the additive <topological group> $\mathbb R$ acting trivially on $\mathbb R$, the continuous one-cocycles are exactly $c(x)=ax$. Additivity first determines values on the <rational numbers>, and continuity extends that identity to all real inputs. All one-coboundaries vanish, giving $H_c^1(\mathbb R,\mathbb R)\cong\mathbb R$. Replacing the coefficients by discrete $\mathbb Z$ gives zero, since a continuous map from the connected real line to a discrete group is constant.
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