Hartogs's theorem on separate holomorphicity

Remarks

If $f$ is also assumed continuous, the result is easier: it is Osgood's lemma. The same easy argument works if $f$ is only assumed locally bounded. In that case the iterated Cauchy formula on small polydiscs gives the power series of $f$. Hartogs's proof first finds, by a Baire category argument, small polydiscs on which $f$ is bounded. It then uses Hartogs's lemma for subharmonic functions on the power series in one variable to get holomorphy at every point. No such result holds for real variables. The function $g(x, y) = xy/(x^2 + y^2)$ with $g(0, 0) = 0$ is real analytic in each variable separately, yet it is not continuous at the origin. Example: $f(z_1, z_2) = e^{z_1 z_2}$ is entire in $z_1$ for each fixed $z_2$ and in $z_2$ for each fixed $z_1$, so it is holomorphic on $\C^2$.

Going deeper

For a student

The theorem shows that the definition of holomorphy in several variables is very robust. Complex differentiability, holomorphy in each variable, local power series expansion, and the Cauchy–Riemann equations for $C^1$ functions all give the same class of functions. In practice one often uses the easier Osgood version, since continuity is usually known. The full theorem is a striking early result of the theory, and its proof introduced methods that reappear in the study of plurisubharmonic functions.