holomorphic function defined on $\mathbb{C}^n$

Question

It is well known that if $D \subset \mathbb{C}$ is an domain then $f:D \to \mathbb{C}$ holomorphic implies $f \in C^{\infty}$. My question is, this property is still true for holomorphic functions $f: D \subseteq \mathbb{C}^n \to \mathbb{C}$ for $n>1$ ?

Answer 1

Yes, holomorphic functions are infinitely differentiable. In fact, in several variables, all you need to assume is that a function is holomorphic in each variable separately and you automatically get that the function is in fact infinitely differentiable (though that is a tad hard to prove). It is not hard to prove that if a function is locally bounded, and holomorphic in each variable separately, then we can iteratively apply the Cauchy formula in each variable separately and from this, as in one variable, obtain that such a function is analytic, that is equal to a power series at each point. So much better than infinitely differentiable in fact.

Usually the definition of holomorphic in several variables is therefore "locally bounded and holomorphic in each variable separately".

Note that such a thing is not true for functions of a real variable. Take $f(x,y) = \frac{xy}{x^2+y^2}$, where you also define $f(0,0)=0$. That function is real-analytic in each variable separately (hold one variable constant). The partial derivatives in $x$ and $y$ exist at all points too. But as a function of two real variables, this function is not even continuous.

See e.g. https://www.jirka.org/scv/ for a bit more.