Identity theorem for holomorphic functions

Given: A complex domain $\mathrm{\Omega }$, holomorphic functions $f,g:\mathrm{\Omega }\to \mathbb{C}$, an accumulation point $a\in \mathrm{\Omega }$ of their agreement set, and the holomorphic difference $h:=f-g$ supplied by Linearity, product, reciprocal, and quotient rules for complex derivatives. A nonempty subset of a connected space that is both open and closed is the whole space (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

For a holomorphic function $h$ on an open set $U$, the set of points having a neighbourhood on which $h$ vanishes is both open and closed in $U$ (The locally zero locus of a holomorphic function is clopen).

A holomorphic function has finite order $m$ at $a$ exactly when it factors near $a$ as $(z-a{)}^{m}q(z)$ with $q(a)\mathrm{\ne }0$; its order is $+\mathrm{\infty }$ exactly when it vanishes on a neighbourhood of $a$ (The order of a zero is the exponent in its local holomorphic factorization).

If $f:U\to \mathbb{C}$ is complex differentiable at $a\in U$, then $f$ is continuous at $a$ (Complex differentiability at a point implies continuity there).