An example of a group with a topology

Take, for instance, $(\mathbb{Q},+)$, endowed with the Zariski topology (that is, a non-empty set $A$ is open if and only if $A^\complement$ is finite). Then the inversion ($x\mapsto-x$) is clearly continuous and addition is clearly separately continuous. But it is not jointly continuous since, for instance $\{(x,y)\in\mathbb{Q}^2\,|\,x+y=0\}$ is not a closed set.

Let $G$ be any infinite group and give it the cofinite topology. Then the product is separately continuous as is inversion, since any bijection $G\to G$ is continuous. But the product is not jointly continuous, since $\{1\}$ is closed but its preimage is not. (Or, you can just cite the fact that any $T_0$ topological group is Hausdorff, so $G$ cannot be a topological group.)