Function holomorphic at a point

This article defines a property that can be evaluated for a function on a (particular kind of) set, and a point in that set. A function satisfying the property at every point, it is termed a holomorphic function
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Definition

In one dimension

Let $U$ be an open subset (without loss of generality, an open connected subset, i.e. a domain) in $\mathbb {C}$. Let $f:U\to \mathbb {C}$ be a function, and let $z_{0}\in U$. We say that $f$ is holomorphic at $z_{0}$ if there exists an open subset $V\ni z_{0}$ such that $f$ is complex-differentiable for every $z\in V$, and further, if the complex differential $f'$ is a continuous function on $V$.