Let $X$ and $Y$ be complex manifolds, and let $F:X\to Y$ be a continuous map. Suppose that for every point $p\in X$ there exist a complex coordinate chart $(U,\varphi)$ on $X$ with $p\in U$, a complex coordinate chart $(V,\psi)$ on $Y$ with $F(p)\in V$, such that $F(U)\subset V$, and the coordinate representative $\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V)$ is holomorphic. Then $F$ is holomorphic as a map of complex manifolds.