I am confused about the property of holomorphism of complex functions.
Take the function $\frac{1}{z}$ as an example. This function satisfies the Cauchy-Riemann equations and is thus said to be holomorphic. However, doesn't the non-analytic point at $z=0$ imply that it is not holomorphic at this point? In that case, does satisfying the Cauchy-Riemann equations simply imply that the function is holomorphic in some sub-domain of the complex plane? And the property of being entire, does that then imply that it satisfies the Cauchy-Riemann equations and is differentiable on the entire complex plane?
A function is not an analytic expression. Its definition requires a domain and a codomain.When we talk about the function $\frac1z$ in the context of Complex Analysis, what we (usually) mean is the function$$\begin{array}{ccc}\mathbb C\setminus\{0\}&\longrightarrow&\mathbb C\\z&\mapsto&\dfrac1z.\end{array}$$Note that $0$ does not belong to its domain. Therefore, there is no problem with $0$. And, yes, it satisfies the Cauchy-Riemman equations at every point of its domain and it is a holomorphic function.
And this function is not entire, since its domain is not $\mathbb C$.