Gateaux complex differentiability and continuity

As is known, there are everywhere discontinuous infinitely Fréchet differentiable functions on the real locally convex spaces $\mathcal D(\mathbb R)$ and $\mathcal D'(\mathbb R)$ of finitely supported infinitely differentiable functions and, respectively, of generalized functions. In this paper the relationship between the complex differentiability and continuity of a function on a complex locally convex space is considered. We describe a class of complex locally convex spaces, which includes the complex space $\mathcal D'(\mathbb R)$, such that every Gateaux complex-differentiable function on a space of this class is continuous. We also describe another class of locally convex spaces, which includes the complex space $\mathcal D(\mathbb R)$, such that on every space of this class there is an everywhere discontinuous infinitely Fréchet complex-differentiable function whose derivatives are continuous.