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---
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 %}
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