The group of continuous homomorphisms

Question

Consider the topological group $\Bbb Q$ with the subspace topology and $\Bbb Z$ with the discrete topology. Is there a characterization for $C^{1}(\Bbb Q,\Bbb Z)$, the group of continuous homomorphisms of $\Bbb Q$ to $\Bbb Z$ with the compact-open topology?

Answer 1

The only group homomorphism $f:\mathbb Q \to \mathbb Z$, continuous or not, is the zero homomorphism.

Proof
If $f$ were non-zero there would exist $q\in \mathbb Q$ with $b=f(q)\neq 0\in \mathbb Z$.
But then we would have for all $0\neq n\in \mathbb N$: $$b=f(q) =f(n\cdot \frac qn)=n\cdot f( \frac qn)$$ so that $f( \frac qn)=\frac bn$.
But this is impossible because $\frac bn \notin \mathbb Z$ for $n$ large enough.