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Suppose we have $\phi(z,w)$ a function of two complex variables and that for each fixed $w$ the function $z\mapsto \phi(z,w)$ is holomorphic, that is, it exists $\frac{\partial}{\partial z}\phi(z,w)$. Is $\frac{\partial}{\partial z}\phi(z,w)$ continous? I know that by Osgood lemma that if both $\frac{\partial}{\partial z}\phi(z,w)$ and $\frac{\partial}{\partial w}\phi(z,w)$ exist then the answer is yes, but if only one of them exist?

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It depends on what you mean by $\frac{\partial f}{\partial z}$ existing, since there are really two interpretations of that. A common one, when applied to smooth functions, is the Wirtinger derivative, but that's probably not what you mean. If you mean that the limits $$ \lim_{h \to 0} \frac{f(z+h,w)-f(z,w)}{h} \qquad \text{and} \qquad \lim_{h \to 0} \frac{f(z,w+h)-f(z,w)}{h} $$ exist at all $(z,w)$, then yes, this means that the function is holomorphic and holomorphic functions are in fact analytic, that is, given by a power series in several variables at all points. The derivatives are also analytic (and hence also holomorphic) and so definitely continuous.

See https://www.jirka.org/scv/ for a short intro that should also explain this particular point, and what Wirtinger operators are.

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