I also made this post in MSE, but I think it may fit here as well.

*Motivation*: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function from $X$ to $Y$ that coincides with $f$ in a dense set, then $g=f$.

Take an analytic function $f: \mathbb{C} \rightarrow \mathbb{C}$. If $g$ is an analytic function that coincides with $f$ in a set with limit point, then $g=f$.

Okay, now given the motivation, my question is the following:

**Is there a theory which has the objective of characterizing functions that satisfy a certain "unique extension" property, in a general context?**

For example, we could define:

Definition: Given a family $\mathcal{F} $ of maps $f:X \rightarrow Y$ , $f \in \mathcal{F}$ is said to satisfy

$P$-unique-extendability on $(X,Y)$if for every subset $A \subset X$ which satisfies the property $P$, we have that every function $g \in \mathcal{F}$ which coincides with $f$ in $A$ is such that $f=g$.

Therefore, for example, every continuous function is *dense-unique-extendable on $(\mathbb{R}$, $\mathbb{R})$*

Maybe there is a categorical POV of this?