Functions with a special property

Question

I was working on a subject. I encountered to the following functions. I was wondering is there a name for such function or property. $$f: A\subseteq \mathbb{R}^2 \longrightarrow \mathbb{R}$$ $\exists \varepsilon $ such that $\forall (x,y)\in A, \exists (x',y')\in A$ such that $(x,y)\neq(x',y')$ and $|f(x,y)-f(x',y')|< \varepsilon$. Thanks to everyone for the help.

Answer 1

A lot of functions satisfy your requirement. A least:

• Bounded ones
• Locally bounded ones
• Continuous ones

I know that it doesn't unfortunately answer your question. But I don't know a special name for functions satisfying your criteria...