Function holomorphic at a point

From Companal

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
View other properties of functions at points

Definition

In one dimension

Let be an open subset (without loss of generality, an open connected subset, i.e. a domain) in . Let be a function, and let . We say that is holomorphic at if there exists an open subset such that is complex-differentiable for every , and further, if the complex differential is a continuous function on .

It turns out that this is equivalent to a function complex-analytic at a point.