Let $f(z,w)$ be defined on $\Bbb{C}\times\Bbb{C}$. Assume for each fixed $z$, $f(z,w)$ is analytic for all $w$. Likewise, assume for each fixed $w$, $f(z,w)$ is analytic for all $z$.
Is it the case that the mixed partial derivatives must exist? That is, WLOG, for a fixed $w_0$, will $$\left.\frac{\partial}{\partial w}f(z,w)\right|_{w=w_0}$$ be analytic for all $z$? If not, what is the counterexample?
Yes, this is a famous theorem of Hartogs: If $f$ is separately holomorphic in $z,w,$ then $f$ is locally representable by power series in $z,w.$