Group with topology which is not topological group

Question

What will example of a group $G$ with topology $\tau$ such that $f: G \to G$ such that $f(x) =x^{-1}$ and $g: G \times G \to G$ such that $g((x,y)) = x * y$ (where $*$ is binary operation on $G$) both are not continuous.

Answer 1

Take the group $\mathbb{Z}/3\mathbb{Z} = \{0,1,2\}$ under addition, with the topology consisting of $\emptyset$, $\{0,1,2\}$ and $\{1\}$.

Now negation is not continuous, since the preimage of the open set $\{1\}$ is $\{2\}$ which is not open.

Similarly, addition is not continuous, since the preimage of the open set $\{1\}$ is $\{(0,1),(1,0),(2,2)\}$ which is again not open in the product topology.

Answer 2

$(\Bbb R,+)$ with the topology generated by the set $[1,\infty)$. In that case the unitary operation is not continuous.

Answer 3

$(\mathbb R,+)$ with the Zariski topology.

If $g:\mathbb R\times \mathbb R\to \mathbb R$ is given by $g(x,y)=x+y$, then the preimage of, say, $(0,1)$ is the set $-x<y<-x+1$ which is not open for the product of the zariski topologies.