[proofplan] We prove the result by checking the definition of a holomorphic map between complex manifolds at an arbitrary point. The hypothesis already supplies, near each point $p\in X$, a pair of complex charts whose coordinate representative for $F$ is holomorphic. Since $F$ is continuous by assumption, these local coordinate representatives are exactly the data required by the definition, so $F$ is holomorphic on all of $X$. [/proofplan] [step:Fix an arbitrary point and choose the chart data supplied by the hypothesis] Let $p\in X$ be arbitrary. By the hypothesis, there exist a complex coordinate chart $(U,\varphi)$ on $X$ with $p\in U$ and a complex coordinate chart $(V,\psi)$ on $Y$ with $F(p)\in V$ such that $F(U)\subset V$ and the map $\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V)$ is holomorphic. [guided] We begin at a single point because holomorphicity of a map between complex manifolds is a local condition on the source. Let $p\in X$ be arbitrary. The assumption of the theorem applies to this point $p$, so it gives an open neighbourhood $U\subset X$ of $p$ equipped with a complex chart $\varphi:U\to\varphi(U)$ and an open neighbourhood $V\subset Y$ of $F(p)$ equipped with a complex chart $\psi:V\to\psi(V)$. The containment $F(U)\subset V$ is important because it makes the coordinate expression meaningful on all of $\varphi(U)$. Indeed, since $\varphi:U\to\varphi(U)$ is a chart, its inverse $\varphi^{-1}:\varphi(U)\to U$ is defined. Since $F(U)\subset V$, the composition $F\circ\varphi^{-1}:\varphi(U)\to V$ is defined, and then applying $\psi:V\to\psi(V)$ gives the coordinate representative $\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V)$. The hypothesis states precisely that this coordinate representative is holomorphic. [/guided] [/step] [step:Compare the obtained coordinate representative with the definition of holomorphicity] By definition, a continuous map $F:X\to Y$ between complex manifolds is holomorphic if for every $p\in X$ there are complex charts $(U,\varphi)$ around $p$ and $(V,\psi)$ around $F(p)$ with $F(U)\subset V$ such that $\psi\circ F\circ\varphi^{-1}:\varphi(U)\to\psi(V)$ is holomorphic. The map $F$ is continuous by hypothesis, and the preceding step verifies the required chart condition at the arbitrary point $p$. [/step] [step:Conclude holomorphicity on all of $X$] Since $p\in X$ was arbitrary, the defining local chart condition holds at every point of $X$. Together with the assumed continuity of $F:X\to Y$, this proves that $F$ is holomorphic as a map of complex manifolds. [/step]