I wanted to use Identity Theorem to verify this functional equation
$$\frac{\beta(1-s)}{\Gamma(s)}=\left(\frac{2}{\pi}\right)^{s}\sin(\pi s/2)\beta(s), \quad \Re(s)\leq 0\tag*{(1)}$$ where $\beta(\cdot)$ represents Dirichlet's Beta Function. I don't have a strong foundation in complex analysis which is why I'm getting lost in the details and would like some help in making sense of how to apply the theorem.
I can identify a subset of the domain $\mathcal{D}$ for which the two sides of the equation are equal. Namely, at the simple poles of $\Gamma(s)$ for negative integers. On the RHS the sine term vanishes for negative even terms and we assume that $\beta(s)$ vanishes for negative odd terms. So, there is a subset $\mathcal{S}=\{s\in\mathcal{D}\,|\,f(s)=g(s)=0\}$ where $f$ and $g$ are the LHS and RHS of $(1)$, respectively. However, the set $\mathcal{S}$ is not open because it is made up of isolated negative integers $s\in\mathbb{N}^{-},V_{\epsilon}(s)\notin\mathbb{N}^{-}$. The condition for the identity theorem is that $\mathcal{S}$ be an open subset of $\mathcal{D}$, which is in turn a connected open subset of the reals. So, I'm guessing I just don't really understand what an open and closed set is to begin with. Next, how do we determine that there is a non-isolated (limit) point within $\mathcal{S}$? Cheers for the help, and apologies for my lack of knowledge in this area.