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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 %}