types of functions between metric spaces Let $X,Y$ be metric spaces (by metric space I mean $(X,d)$ with $d:X\times X\to [0,\infty ]$ satisfying the triangle inequality, symmetry and $d(x,x)=0$). The following definitions for a function $f:X\to Y$ are well-known:
1) $f$ is nonexpanding if $d(f(x),f(x'))\le d(x,x')$. 
2) $f$ is Lipschitz if there exist a constant $C>0$ such that $d(f(x),f(x')) < C\cdot d(x,x')$. 
3) $f$ is uniformly continuous if for all $\epsilon >0 $ there is $\delta >0$ such that if $d(x,x')< \delta $ then $d(f(x),f(x'))< \epsilon $.
4) $f$ is continuous if for all $x\in X$ and $\epsilon > 0$ there is $\delta >0$ such that if $d(x,x')< \delta $ then $d(f(x),f(x'))< \epsilon$.
5) $f$ is large scale Lipschitz if there exist constants $C,A$ such that $d(f(x),f(x')) < C\cdot d(x,x') + A$.
6) $f$ is bornologous if for all $R>0$ there is $S>0$ such that if $d(x,x')<R$ then $d(f(x),f(x')) < S$.
7) $f$ is weak bornologous if there exist $R,S>0$ such that if $d(x,x')<R$ then $d(f(x),f(x')) < S$. 
Clearly $1\implies 2 \implies 3 \implies 4$ and $5\implies 6 \implies 7$. 
A definition I did not find written anywhere is the following. First an auxiliary concept: a function $\alpha : [0,\infty ]\to [0,\infty ]$ satisfying $\alpha (0)=0$, $\alpha$ is nondecreasing, $\alpha (s+t)\le \alpha (s)+\alpha (t)$, and for all $\epsilon >0$ there is $\delta >0 $ such that $\alpha (\delta )<\epsilon $ shall be called a distance distortion function. Any distance distortion function associates with any metric space $X$ a new metric space, $\alpha _*(X)$, with the same underlying set, and with distance function $\alpha_*(d):X\times X\to [0,\infty ]$, given by $\alpha _*(X)(x,x')=\alpha (d(x,x'))$. 
2.5) say that $f$ is relatively nonexpanding if there exists a distance distortion function $\alpha$ such that $f:\alpha _*(X)\to Y$ is nonexpanding. 
It is clear that $2\implies 2.5 \implies 3$.
So my questions are:


*

*Does the concept given by definition 2.5 have a name, if so, any references will be greatly appreciated.

*Is the list of concepts above a complete list of notions for structure preserving functions between metric spaces, for sensible notions of structure (e.g., above, rigid metric preservation, weaker metric preservations, uniform preservation, continuity, and the large scale notions).
Thanks!  

Later addition: Condition 2.5 is known as a function admitting a special modulus of continuity, in particular subadditive/concave (thanks @5pm for that information, thus answering my first question). 
 A: More remarks about (2.5): 


*

*$(2.5) \iff (3) + (6)$. That is, (2.5) describes bornologous uniformly continuous maps. 

*A comment on "special" moduli of continuity. The only really special property is being finite. As 
long as there is a finite modulus of continuity, we can take the concave majorant of it, which is also finite.  

*(2.5) can be rephrased once more as: for every $\epsilon>0$ there exists $\delta>0$ such that (... as in uniform continuity), and additionally the supremum of allowed $\delta$ tends to infinity as $\epsilon\to \infty$.  This form has some shock value: $\epsilon\to\infty$. 



For the second question. There are also "two-sided" notions that require invertibility and having the inverse with the same property. 
Isometries,  bi-Lipschitz maps, and uniform homeomorphisms are examples of these. Then there are 
quasisymmetric maps (which do not admit a description by 
a pair of one-sided conditions); even more general quasi-Möbius maps, quasi-isometries (not to be confused with bi-Lipschitz maps), and maps preserving other coarse structures. 
And probably a few more that were invented by someone as I was typing this.
