 Solution

For two moving vertices $p(s),q(s)$, the integral along their joining segment isThe integral factor is continuous in $s$ by the allowed fact, and the other factor is continuous because the vertices are continuous. Thus $J_{p,q}$ is continuous, including when its two endpoints coincide.

The integral around the triangle is exactlyEach term is continuous, so their sum is continuous at zero andNo holomorphicity is used here. Compactness of the fixed containing triangle also gives a direct proof of the allowed fact through uniform continuity of $f$.