The author shows: Let \(E\) be a complex vector space \(E\) endowed with its finest locally convex topology. Then there is a discontinuous complex valued polynomial on \(E\) iff the algebraic dimension of \(E\) is uncountable (Proposition 2). A lemma (Lemma 1) gives the possibility to join the proof of Lemma 19 in [J. A. Barroso, M. C. Matos and L. Nachbin, Infin. Dim. Holom. Appl., Proc. Int. Symp., Univ. Estadual de Campinas/Brasil 1975, 31-74 (1977; Zbl 0399.46034)].
Now, the author is able to prove Proposition 3:
Let \(E\) be a Hausdorff complex locally convex space, \(U\subseteq E\) an open subset, and \(F\) a complex locally convex space. Then any finitely holomorphic map \(f:E\to F\) (i.e. the restriction of \(f\) to \(U\cap S\) is holomorphic for every finite dimensional vector subspace \(S\) of \(E\)) is holomorphic iff \(E\) is countable dimensional and carries its finest locally convex topology.