Gateaux complex differentiability and continuity

Research output: Contribution to journalArticlepeer-review


As is known, there are everywhere discontinuous infinitely Frechet differentiable functions on the real locally convex spaces D(R) and V(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 V(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 D(R), such that on every space of this class there is an everywhere discontinuous infinitely Frechet complex-differentiable function whose derivatives are continuous.
Original languageEnglish
Pages (from-to)1217-1227
Number of pages11
JournalIzvestiya. Mathematics
Issue number6
Publication statusPublished - Nov 2004

ASJC Scopus subject areas

  • Mathematics(all)


Dive into the research topics of 'Gateaux complex differentiability and continuity'. Together they form a unique fingerprint.

Cite this