= Solution
Only the right-hand ball is <closed>, but the union is still \b[<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 $0\leq t<1$, <convexity> and the <triangle inequality> give
$$
\lVert(1-t)z-a\rVert
\leq(1-t)\lVert z-a\rVert+t\lVert a\rVert<1.
$$
At $t=1$, the <path> reaches zero in the other, closed ball. Points in that ball also join zero by <line segments>, completing the construction.
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