I’ve been reading through Section 9.2 in Conways book on complex functions and I am a bit stuck on understanding Conway’s remark in the second to last paragraph on p.214. To be precise,
“That is, $$f_s(z) = f_t(z), z \in D_s \cap D_t$$ whenever $|s-t| < \delta$.”
This doesn’t appear to be the case based on (b) of Definition 2.2 unless the intersection $D_s \cap D_t$ is also connected. I see that $D_s$ and $D_t$ must be regions (open, connected sets) but this doesn’t ensure that their intersection is connected. I can see that this is true for the connected component containing $\gamma (s)$. Is it the case that I’m just missing something and that we don’t need their intersection to be connected?