Pointwise bounded family

A family $\mathcal{F}$ of functions $f:X\to\mathbb{R}$ is pointwise bounded if for every $x\in X$ the set of values $\{f(x): f\in\mathcal{F}\}$ is bounded in $\mathbb{R}$, equivalently

Pointwise boundedness is weaker than being uniformly bounded (which requires one bound to work for all $x$ at once). Together with equicontinuity hypotheses, it appears in compactness results for subsets of spaces of continuous functions such as Arzelà–Ascoli .

Examples:

• On $[0,1]$, the family $\{f_n\}_{n\ge 1}$ with $f_n(x)=x^n$ is pointwise bounded since $0\le f_n(x)\le 1$ for all $x$ and $n$.
• On $[0,1]$, the family $f_n(x)=n x$ is not pointwise bounded (for any fixed $x>0$, the values $n x$ are unbounded as $n\to\infty$).