  Examples of differentiable mappings into real or complex topological vector
spaces with specific properties are given, which illustrate the differences
between differential calculus in the locally convex and the non-locally convex
case. In particular, for a suitable non-locally convex space E, we describe a
smooth injection of R into E whose derivative vanishes identically; we present
a complex C^\infty-map on the complex field C which is not given locally by its
Taylor series, around any point; we present a complex C^1-map into a complete,
non-locally convex topological vector space which is not C^2; and we present a
compactly supported, non-zero, complex C^\infty-map from C to a suitable
non-locally convex space.
