I am reading the definition of equicontinuous family at a point from book called "Iteration of Rational Functions" by Alan F. Beardon. There it is written that equicontinuity of family at a point is the formal expression for the idea of "preservation of proximity".
I didn't understand what does preservation of proximity mean? It will be better if someone explains this by example.
For a continuous function $f$, we maintain proximity to $f(x_0)$ as long as $x$ is in some neighbourhood $N(x_0)$ of $x_0$. By proximity, I mean that $\lvert f(x) - f(x_0)\rvert < \varepsilon$.
However, if I consider a different continuous function $g$, I might not maintain the same proximity of $g$ to $g(x_0)$ when $x$ is in $N(x_0)$.
If $f, g$ are members of an equicontinuous family, I can maintain this proximity. That is, I can guarantee that $g$ is near (at most $\varepsilon$-away) $g(x_0)$ for the same neighbourhood $N(x_0)$.
You can try working out the following examples if it helps:
Example of equicontinuous family (at any point in $\mathbb{R}$): $\{x\mapsto x+n : n\in\mathbb{N}\}$.
Example of non-equicontinuous family (at any point in $\mathbb{R}$): $\{x\mapsto nx : n\in\mathbb{N}\}$.