---
title: Basics
parent: One Complex Variable
grand_parent: Complex Analysis
nav_order: 1
# cspell:words
---

# {{ page.title }}

{: .theorem }
> {: #holomorphic-function-is-constant-if-derivative-vanishes }
>
> If the derivative of a holomorphic function vanishes
> throughout a connected open subset of the complex plane,
> then it must be constant on that set.
>
> More generally, if the derivative of a holomorphic function vanishes
> throughout an open subset of the complex plane,
> then it must be constant on any connected component of that set.

{% proof %}
{% endproof %}