Let $U\subseteq \mathbb{C}$ be an open subset, and $f:U\to \hat{\mathbb{C}}$ be a continuous function. Let $\mathrm{\Delta }\subseteq U$ be a triangle contained in $U$. Suppose that for any choice of $\mathrm{\Delta }$, we have

Then $f$ is 1;; holomorphic on $U$.