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.
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