# 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).

• 2.5) says that $f$ admits a global modulus of continuity. It is equivalent to having a pointwise finite function $\alpha$ (with limit $0$ at $0$) such that $d_Y(f(a),f(b))\le \alpha(d_X(a,b))$ for all $a,b$. The concavity and monotonicity can be arranged by taking the concave majorant. – user53153 Mar 9 '13 at 5:00
• thaks @5pm. Any good reference you can recommend or shall I just start googling? – Ittay Weiss Mar 9 '13 at 5:04
• The Wikipedia article on modulus of continuity calls them "special uniformly continuous functions". Yes, googling is probably the way to go. – user53153 Mar 9 '13 at 5:14
• Yes, I see. Thanks again. – Ittay Weiss Mar 9 '13 at 5:19

• $(2.5) \iff (3) + (6)$. That is, (2.5) describes bornologous uniformly continuous maps.
• (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: $\epsilon\to\infty$.