 Strengthened Weierstrass condition

For a $C^1$ curve $u$, some $\delta >0$ must make the Weierstrass excess function nonnegative for every $x\in [a,b]$, every trial slope $q\in \mathbb{R}$, and all $y,p$ satisfying $|y-u(x)|<\delta$, $|p-u^{\prime }(x)|<\delta$. Thus the supporting-line property persists in a tube of nearby heights and reference slopes. The term “strengthened” here refers to this neighborhood requirement; strict excess for $q\ne p$ is a further condition.