pages/complex-analysis/one-complex-variable/basics.md


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

# {{ page.title }}

{% theorem %}
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.
{% endtheorem %}

{% proof %}
{% endproof %}