Is the multiplicity of the zero of a holomorphic function is constant?

Question

Let $\Omega$ be a domain in $\mathbb{C}^{n},n\geq 2$ and $f:\Omega\mapsto\mathbb{C}$ be a holomorphic function. Define $ord_{a}f$ to be the total order of f at the point $a$ (also called the multiplicity of the zero of the function $f$ at the point $a$). If the zero set of $f$, namely $Z(f)$ is assumed to be non-empty, connected and irreducible, then is it true that $ord_{z}f$ is constant $\forall z\in Z(f)~?$