French Railroad Metric/Topology
For any $x \in \Bbb R^2$ with $x\neq 0$ we have that $\{x\}$ is open and is in any base for $\tau$ while basic neighbourhoods of $0$ are the standard Euclidean ones, $B_r(0)= \{y\mid \|y\| < r\}$ for $r>0$.
Consider what this means for continuity...