Let $f: X_1 \rightarrow X_2 $ be a function between two metric spaces. My question is, if there is a name in the standard literature for the following property of $f$ in $x \in X_1$:

$$(1)~~~~~~~~~~~ \forall \varepsilon > 0 ~~\exists \delta > 0 :~~\operatorname{im}_f(B_\varepsilon(x) ) \subseteq B_\delta(f(x))$$ i.e. the image of every $\varepsilon$-ball is bounded.

By looking for obvious candidates for a name, I only found the term of ,,local boundedness in $x$'' which can in this case be expressed as $$(2)~~~~~~~~~~~ \exists \varepsilon > 0 ~~\exists \delta > 0 :~~\operatorname{im}_{\vert f \vert}(B_\varepsilon(x) ) \subseteq B_\delta(0)$$ They are of course not equivalent, since the for example: the real function $f(x)=x^{-1}$ is locally bounded in every point $x \neq 0$ but does not have property $(1)$ in any point.

  • $\begingroup$ Is the $\delta$ mentioned global? That is, is the condition $\forall\epsilon>0,\,\exists\delta>0,\,\forall x\in X_{1}\ldots$ the condition you are intending, or does that $\delta$ depend on $x$? $\endgroup$ – Robert Thingum Mar 31 '19 at 0:03
  • $\begingroup$ A consequence of your definition (regardless of whether or not the $\delta$ is global) is that if $f:X_{1}\rightarrow X_{2}$ is such a map then $f(X_{1})\subseteq B(f(x),\delta_{x})$ for every $x\in X_{1}$. Not sure if you want this or not. $\endgroup$ – Robert Thingum Mar 31 '19 at 0:06
  • $\begingroup$ @RobertThingum No, $\delta$ is not meant to be global. So the quantification was more intended to be $\forall x \in X_1 ~\forall \varepsilon ~\exists \delta ~...$, meanig $\delta = \delta(\varepsilon,x)$. $\endgroup$ – Nemo Mar 31 '19 at 12:24
  • $\begingroup$ @RobertThingum I think $f(X_1) \subseteq B(f(x) , \delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(\mathbb{R} \setminus \{0\} ) = \mathbb{R} \setminus \{0\} $. $\endgroup$ – Nemo Mar 31 '19 at 12:30
  • $\begingroup$ Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps $\endgroup$ – Robert Thingum Mar 31 '19 at 13:29

The functions satisfying $(1)$ are the functions sending metrically bounded sets of $X_{1}$ to metrically bounded sets of $X_{2}$.

See the following

Bornological Space on wiki

If you are interesting in reading about some applications of these kinds of functions within an active area of research you should look into coarse geometry.

Coarse Spaces on wiki

  • $\begingroup$ Is it right to say $f$ preserves metrically bounded sets? $\endgroup$ – Alex Vong Mar 31 '19 at 13:44
  • $\begingroup$ I would say so. You could also say that $f$ simply preserves "boundedness". $\endgroup$ – Robert Thingum Mar 31 '19 at 13:45

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