Let $G=\mathbb C\setminus\{x+iy:x\leq0,y=0\}$ and $H=\mathbb C\setminus\{x+iy:x\in\mathbb Z,x\leq0,y=0\}$. Suppose f:G→C and g:H→C are analytic. I: $\int_\gamma f\,dz$ is independent of paths in G joining −i and i. II: $\int_\gamma g\,dz$ is independent of such paths in H.
G is the slit plane and is simply connected, so every analytic f on G has a primitive and I is true. H instead has isolated deleted points, including zero. The function g(z) = 1/z is analytic on H, but adding a loop around zero changes its path integral by $2\pi i$. Thus II need not hold and B is correct.
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