Suppose a function $f: \mathbb{R} \to \mathbb{R} $ continuous only for $x \in \mathbb{N}$

Is there some terminology to say if the continuity is "uniform" on $x \in \mathbb{N}$?

Clearly uniform continuity doesn't apply on $\mathbb{R}$.

Moreover, uniform continuity is trivially true for $f: \mathbb{N} \to \mathbb{R}$ since $\mathbb{N}$ is comprised of isolated points.

Hence I suggest an other type of uniform continuity,

(D?) A function $f: X \to Y$ is uniformly continuous on the neighboorhood of a set $A \subseteq X$ if for any $\epsilon > 0$ there exists a $\delta > 0$ such that for $x \in A$ and $y \in X$,

$d_X(x,y) < \delta \implies d_Y(f(x) - f(y)) < \epsilon$

For example, the function

$ f(x)=\begin{cases} \sin\pi x,&\text{if }x\in\Bbb Q\\ 0,&\text{if }x\in\Bbb R\setminus\Bbb Q\;. \end{cases} $

is uniformly continuous on the neighborhood of $\mathbb{N}$ and not continuous on $\mathbb{R} \setminus \mathbb{N}$. However, the related function

$ g(x)=\begin{cases} \sin\pi x \times [x-.5],&\text{if }x\in\Bbb Q\\ 0,&\text{if }x\in\Bbb R\setminus\Bbb Q\;. \end{cases} $

with $[z]$ the integer part of $z$ is not uniformly continuous on the neighborhood of $\mathbb{N}$.

Is my proposed definition correct and helpful? Has something similar been used in the literature?


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