I have a question on notation.
Let $\mathcal{F}$ be an equicontinuous family of functions in $C(K)$, where $K$ is a compact subset of $\mathbb{R}^{n}$. Suppose that for each $x \in K$,
$$\sup\{f(x):f \in \mathcal{F} \} = M_{x}$$
is finite.
What are a few terms of $\{f(x):f \in \mathcal{F}\}$ supposed to look like? Do we take an arbitrary function and then find the supremum of $f(x)$ over all $x$ in $K$?