Complex-valued continuous function gives closed form iff holomorphic

This fact relates notions of complex analysis and complex differentiation with de Rham cohomology.
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Statement

Suppose $U\subset \mathbb{C}$ is an open subset and $f:U\to \mathbb{C}$ is a continuous function. Consider the differential 1-form associated to $f$:

$f(z)dz$

Then, this 1-form is closed if and only if $f$ is a holomorphic function.