 Morera gluing lemma across a straight line

Let $U\subseteq \mathbb{C}$ be open, let $L$ be a straight line, and let $f:U\to \mathbb{C}$ be continuous on $U$ and holomorphic on $U\setminus L$. Then $f$ is holomorphic on all of $U$. Near a point of $L$, work in a disk whose closure lies in $U$. Split a small filled triangle along $L$ and triangulate the two resulting convex polygons. Translate each triangle slightly into its corresponding open half-plane, keeping it in $U$. Its contour integral is zero by Cauchy's integral theorem. On each segment the integral is the endpoint difference times the integral of $f$ along the unit-interval parametrization; uniform continuity on a containing compact disk therefore makes the contour integral continuous as the vertices move back. Each untranslated piece consequently has zero integral. Internal oriented edges cancel when the pieces are added, so every small triangle has zero integral. Morera's theorem proves the claim locally along the line. The continuity across $L$ is essential; holomorphicity on the two sides alone is insufficient.