Reciprocal of Holomorphic Function

Theorem
Let $f: \C \to \C$ be a complex function.

Let $U \subseteq \C$ be an open set such that $f$ has no zeros in $U$.

Suppose further that $f$ is holomorphic in $U$.

Then the complex function
 * $\dfrac 1 {f_{\restriction U} } : U \to \C$

is holomorphic.

Proof
Let $g: U \to \C$ be such that $\map g x = 1 / \map f x$.

Since $\map f x$ is nonzero for $x \in U$, $g$ is well-defined.

By Quotient Rule for Continuous Complex Functions, $g$ is continuous.

Let $z_0 \in U$.

As $g$ is continuous:
 * $\ds \lim_{h \mathop \to 0} \frac 1 {\map f {z_0 + h} } = \frac 1 {\map f {z_0} }$

As $f$ is holomorphic:
 * $\ds \lim_{h \mathop \to 0} \frac {\map f {z_0 + h} - \map f {z_0} } h = \map {f'} {z_0}$

By the Combination Theorem for Limits of Complex Functions:

It follows that $g$ is holomorphic.