I was thinking about a map f such that $Graph(f)$ is connected but it is not path connected, this is the map:
$$f(x)=\left\{\begin{array}{cl} \sqrt{x^2-1} &\ x\in \mathbb{Q} \\ -\sqrt{x^2-1} &\ x \in \mathbb{R}-\mathbb{Q} \end{array}\right.$$
It is clear that it is not path connected in each segment that connects two points there are rational and irrational numbers, so my question is:
It is connected?
it is not connected because it can be represented as the union of two disjoint non-empty open sets:
$$A := (-2,2) \times \left(-2,\sqrt{1-\frac{\pi^2}{4^2}}\right)$$
$$B := (-2,2) \times \left(\sqrt{1-\frac{\pi^2}{4^2}},2\right)$$
As $A \cap B = \varnothing$ and $ Graph(f) \subset A \cup B$ so it is not connected.