Past exam of the metric spaces and complex analysis course of the University of Oxford / 2015 / 2 / c / iii

Only the right-hand ball is closed, but the union is still path-connected and connected. Its right-hand closed ball contains zero, and every point $z$ of the left-hand open ball can be joined to zero by $\gamma (t)=(1-t)z$.

To check the endpoint carefully, write $a=(-1,0)$. For $0\le t<1$, convexity and the triangle inequality giveAt $t=1$, the path reaches zero in the other, closed ball. Points in that ball also join zero by line segments, completing the construction.