Function continuously real-differentiable at a point

{at-point function property|continuously real-differentiable function}}

Definition

In one dimension over complex numbers

Let $U\subset \mathbb {C}$ be an open subset (without loss of generality, an open connected subset, i.e. a domain). Let $f:U\to \mathbb {C}$ be a function, and $z_{0}\in U$ be a point. We say that $f$ is continuously real-differentiable at $z_{0}$ if there exists a neighborhood $V\ni z_{0}$ such that $Df$ (the Jacobian of $f$ exists at all $z\in V$ and each of its entries is continuous as a function of $z$.

For maps between real vector spaces

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Relation with other properties

Stronger properties

• Function holomorphic at a point
• Function real-analytic at a point