Is set $$ A = \left\{ (x,mx) \mid m \in \mathbb{N}, x \in [ 0 , 1 ] \right\} \cup \left\{ (x,-mx) \mid m \in \mathbb{N}, x \in (0 , 1) \right\} $$ connected?
I tried by dividing the set into two open sets whose union is the whole set and are disjunct and nonempty, but I can't. The problem is the zero.
Hint: for each $m$ you have a graph of a continuous function from a connected domain. So the set $A$ is a union of a lot of graphs of continuous functions. More than that, note that for each graph of the type $\{(x,-mx):x\in (0,1)\}$ you can add the point $(0,0)$ to the graph because this point is anyway a point in $A$. So that way you get that the intersection of all the graphs in the union is not empty.