Continuity of a function $f$ on a set $A$ appears less strong than $f$ being pointwise continuous on every $x$ on a set $A$.
Specifically, the two definitions differ for points at the boundary of $A$.
More precisely,
(1) A function $f$ with domain $D$ is continuous on a set $A \subseteq D$ if, for any $x$ in $A$ and $\epsilon > 0$, the exists a $\delta > 0$ such that for $\bf y \in A$, $|x - y| < \delta $ implies $|f(x) - f(y)| < \epsilon$, .
(2) A function $f$ with domain $D$ is continuous for every $x \in A$ if, for any $x \in A$ and $\epsilon > 0$, there exists a $\delta > 0$ such that for any $\bf y \in D$, $|x - y| < \delta $ implies $|f(x) - f(y)| < \epsilon$.
That is, for (2), the requirement that $y$ be in $A$ is dropped. So clearly, (2) is stronger than (1) in general.
For open sets, the two definitions are equivalent, in that we can always pick a $\delta$ small enough that such that $|x-y| < \delta$ implies $y$ is in $A$.
My question is the following: is it possible to strengthen the definition of Uniformly Continuity & local Lipschitz continuity on a set $A$ to be more in line with (2)?
That is, something like
(3?) A function $f$ with domain $D$ is pointwise continuous uniformly for every $x \in A$ if for any $\epsilon > 0$, the exists a $\delta > 0$ such that for any $y \in D$ and every $x \in A$, $|x - y| < \delta $ implies $|f(x) - f(y)| < \epsilon$.
(4?) A function $f$ with domain $D$ is locally Lipschitz continuous uniformly for every $x \in A$ if there exists a $K$ such that every $x \in A$, there exist a neighborhood $U_x$ of $x$ in $D$ so that $f$ is Lipschitz continuous on $U_x$ with constant $K_x < K$.
As you can see (3) and (4) are stronger than the usual definitions of uniform continuity on a set and local Lipschitz continuity.
Interestingly, if $A$ is compact, (4) is also stronger than Lipschitz continuity on $A$ since for every $x$ on the boundary of $A$, there also exists a neighborhood not contained in $A$ such that $|f(x) - f(y)| \le K|x - y|$ for $y$ in that neighborhood.
Do definitions similar to (3) and (4) appear in the literature? They would be quite useful for my work, in particular to allow the uniform convergence of $|f(x) - f(y)|$ when $y$ converges from outside $A$ to the boundary of $A$.
Initially, continuity of a function is defined at a single point. Once that definition is under our disposal, we can define "continuity of a function on a set $A$", by requiring that it would be continuous at every point of the set $A$. Therefore, the two notions are precisely equivalent: continuity on a set means exactly this: continuity at every point of the set.