Let $(X, d)$ and $(Y, \rho)$ be compact metric space. If continuous map $f:(X, d)\rightarrow (Y, \rho)$ is Lipschitz map then for every $\epsilon>0$ there is $\delta>0$ such that for every $n\in\mathbb{N}$ when $\sum_{i=1}^{n}d(x_i, y_i)<\delta$ we have $\sum_{i=0}^{n}\rho(f(x_i), f(y_i))<\epsilon$.
In my research I need to find conditions to imply that for every $\epsilon>0$ there is $\delta>0$ such that for every $n\in\mathbb{N}$ when $\frac{\sum_{i=1}^{n}d(x_i, y_i)}{n}<\delta$ we have $\frac{\sum_{i=0}^{n}\rho(f(x_i), f(y_i))}{n}<\epsilon$.
Can you help me to know it.