# Name for functions with certain boundedness property

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.

• 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 $x$? – Robert Thingum Mar 31 '19 at 0:03
• 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. – Robert Thingum Mar 31 '19 at 0:06
• @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 $\delta = \delta(\varepsilon,x)$. – Nemo Mar 31 '19 at 12:24
• @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 $f(\mathbb{R} \setminus \{0\} ) = \mathbb{R} \setminus \{0\}$. – Nemo Mar 31 '19 at 12:30
• Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps – 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 $$X_{2}$$.
• Is it right to say $f$ preserves metrically bounded sets? – Alex Vong Mar 31 '19 at 13:44
• I would say so. You could also say that $f$ simply preserves "boundedness". – Robert Thingum Mar 31 '19 at 13:45