Equicontinuity, pointwise boundedness, and uniform boundedness for families in $C(K,\mathbb R)$

Let $(K,d)$ be a nonempty compact metric space and let $\mathcal F\subseteq C(K,\mathbb R)$ in the sense of The space $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space. The family $\mathcal F$ is equicontinuous at $a\in K$ when, for every $\varepsilon>0$, there is $\delta>0$ such that for every $f\in\mathcal F$ and every $x\in K$, $d(x,a)<\delta$ implies $|f(x)-f(a)|<\varepsilon$. It is equicontinuous when this holds at every $a\in K$.

It is pointwise bounded when, for every $a\in K$, the set $\{f(a):f\in\mathcal F\}$ is bounded in $\mathbb R$. It is uniformly bounded when a real $M\ge0$ satisfies $|f(x)|\le M$ for every $f\in\mathcal F$ and $x\in K$.