Moduli of Continuity of finite family of maps

Question

Could anyone tell me how to define moduli of continuity for a finite family of Lipschitz self-maps on some complete, separable metric space? Thanks for helping.

Answer 1

This is the definition I'm familiar with.

Given a family $\mathcal F$ of functions between two fixed metric spaces $(X,d_X)$ and $(Y,d_Y)$, we say that a function $\rho\colon \mathbf R_+\to \mathbf R_+$ is its modulus of continuity if it has the property that for every $f\in \mathcal F$, every pair $x_1,x_2$ of points in $X$ and every real $\varepsilon\in\mathbf R_+$, we have that if $d_X(x_1,x_2)<\rho(\varepsilon)$, then $d_Y(f(x_1),f(x_2))<\varepsilon$.

This is the only sensible definition that I can think of (well, except for trivial variations, like additional constraints about $\rho$ like monotonicity, or changing the inequality into a non-strict one). Whether $\mathcal F$ is finite or not, and whether $X,Y$ are separable or complete does not matter.