The canonical morphism $\mathbb{R}_{\text{discrete}} \rightarrow \mathbb{R}_{\text{continuous}}$ of Abelian topological groups should induce morphisms $B^n\mathbb{R}_\text{discrete} \rightarrow B^n\mathbb{R}_{\text{continuous}}$, giving rise to a morphism $$ H^n(X;\mathbb{R}_{\text{continuous}}) \rightarrow H^n(X;\mathbb{R}_\text{discrete}) $$ natural in "nice" topological spaces $X$, for each integer $n\geq0$; here we are using the "topological group cohomology" given by $H^n(X;A) = \pi_0\mathrm{Hom}_{\mathrm{Top}}(X,B^nA)$ for $A$ a ("nice") Abelian topological group.
If I'm not mistaken, the RHS $H^n(X;\mathbb{R}_{\mathrm{discrete}})$ should coincide with the "ordinary" singular cohomology with coefficients in the Abelian group $\mathbb{R}$.
I was wondering if there is any nice description of the LHS of the above homomorphism, and the homomorphism itself?
Because $\mathbb{R}$ is contractible, so are all $B^n\mathbb{R}_{\text{continuous}}$. So $H^n(X;\mathbb{R}_{\text{continuous}}) = 0$ for all $X$.