Example for algebraic homomorphism between topological groups which is not continuous I am quite sure there should be an easy example of:

(algebraic) homomorphism between topological groups which is not continuous.

However, I do not see one immediately.
The identity map from the real numbers with the usual topology to the real numbers with the discrete topology.
Consider
$$\begin{align}f\colon\quad\mathbb Q[\sqrt 2]&amp;\to \mathbb Q[\sqrt 2],\\ a+b\sqrt 2&amp;\mapsto a-b\sqrt 2\end{align}$$
where the topology is that inherited from $\Bbb R$.


   Define two sequences of integers as follows: Let $a_1=-1,b_1=1$ and  recursively $a_{n+1}=a_n^2+2b_n^2, b_{n+1}=2a_nb_n$. Then with $x_n:=a_n+b_n\sqrt 2$ we have $x_{n+1}=x_n^2$ and from $|x_1|=\sqrt 2-1&lt;1$ we see that $x_n\to 0$. On the other hand $x_nf(x_n)=a_n^2-2b_n^2\ne 0$ because $\sqrt 2$ is irrational and $a_n=b_n=0$ (i.e., $x_n=0$) does not occur. We conclude $|f(x_n)|=\frac{|a_n^2-2b_n^2|}{|x_n|}\ge \frac1{|x_n|}\to \infty$. If $f$ were continuous, we'd expect $f(x_n)\to f(0)=0$ instead.