Theorem: Whole-cell triangular enclosure

Partition $[0,1]$ into cells $I_j=[t_j,t_{j+1}]$. For $u\in I_j$, split each defect integral into earlier full cells and $[t_j,u]$.

On an earlier cell $I_k$, enclose its integrand using

then multiply by $|I_k|$. On the last partial interval use $u-s\in [0,|I_j|]$ and multiply the integrand rectangle by $[0,|I_j|]$. The resulting defect rectangles cover all $u\in I_j$.

A rectangle $[a,b]+\mathrm{i}[c,d]$ has modulus at most

Taking the maximum over cells produces a uniform $ϵ$.

Proof. The integral average of a function in a convex rectangle remains in that rectangle. The partial interval has length in $[0,|I_j|]$. Interval arithmetic preserves inclusion, and $|x+\mathrm{i}y|\le |x|+|y|$.