Pointwise bounded family

Let $X$ be a set, let $(Y,d_Y)$ be a metric space , and let $\mathcal{F}$ be a family of functions $f:X\to Y$.

The family $\mathcal{F}$ is pointwise bounded if for every $x\in X$ the set of values $\mathcal{F}(x)=\{f(x): f\in\mathcal{F}\}\subseteq Y$ is bounded in $Y$, meaning there exist $y_x\in Y$ and $M_x<\infty$ such that $d_Y\bigl(f(x),y_x\bigr)\le M_x\quad\text{for all } f\in\mathcal{F}.$

For real-valued families $\mathcal{F}\subseteq \mathbb{R}^X$, this reduces to: $\forall x\in X\ \exists M_x<\infty\ \forall f\in\mathcal{F}:\ |f(x)|\le M_x.$

Pointwise boundedness is a natural “local boundedness” hypothesis; on compact domains, combined with equicontinuity it often upgrades to uniform boundedness .

Examples:

• If $\mathcal{F}=\{f_n\}$ with $f_n(x)=x/n$ on $X=\mathbb{R}$, then $\mathcal{F}$ is pointwise bounded (at each fixed $x$, $|x/n|\le |x|$).
• The family $f_n(x)=n$ on any nonempty $X$ is not pointwise bounded.