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Suppose $F \colon \mathbb{C} \times \mathbb{C} \to \mathbb{C}$ is a continuous function of two complex variables which is holomorphic in each variable (i.e. for each $w \in \mathbb{C}$ the functions $z \to F(z,w)$ and $z \to F(w,z)$ are holomorphic). Show that the function $g(z) = F(z,z)$ is holomorphic.

At first I thought to show that partial derivatives of $g$ are continuous and satisfy the Cauchy-Riemann equations. It is easy to see that they satisfy the Cauchy-Riemann equations since $F$ is holomorphic in each variable separately, but I am not sure about the continuity. I don't really know how to use the fact that $F$ is continuous.

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By the Osgood's lemma we have $F$ is holomorphic as function of two complex variable, which means that locally it is given as power series. In particular, $F$ is a smooth function. Therefore, you can calculate $$\frac{\partial}{\partial \overline z} f(z,z) =0$$ and conclude $z \mapsto f(z,z)$ is holomorphic. Just take a look at the Osgood's lemma, it is not a hard result.

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