Uniform big O bound
= Uniform big O bound
{title2=$|f_\alpha(n)|\le C|g(n)|\quad\text{with }C,N\text{ independent of }\alpha$}
A uniform big O bound for a family of <functions> uses one multiplier and one threshold for every allowed parameter:
$$
\exists C>0,\ N\quad\forall\alpha\in A\quad\forall n\ge N:
\quad |f_\alpha(n)|\le C|g(n)|.
$$
The order of the quantifiers matters. For $f_\alpha(n)=\alpha n$, $0\le\alpha\le10$ admits $C=10$ and $N=1$, while the unbounded family $\alpha>0$ admits no such constant. Each individual member nevertheless satisfies $f_\alpha=O(n)$. A uniform bound is useful when an approximation must work across a whole parameter range, rather than separately for each fixed parameter.