Uniform big O bound ($|f_{\alpha }(n)|\le C|g(n)|\text{with }C,N\text{ independent of }\alpha$)

A uniform big O bound for a family of functions uses one multiplier and one threshold for every allowed parameter:The order of the quantifiers matters. For $f_{\alpha }(n)=\alpha n$, $0\le \alpha \le 10$ 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.