  Let $C$ be a convex subset of a locally convex space. We provide optimal
approximate fixed point results for sequentially continuous maps $f\colon
C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an
approximate fixed point net. Next, it is shown that if $C$ is bounded but not
totally bounded, then there is a uniformly continuous map $f\colon C\to C$
without approximate fixed point nets. We also exhibit an example of a
sequentially continuous map defined on a compact convex set with no approximate
fixed point sequence. In contrast, it is observed that every affine
(not-necessarily continuous) self-mapping a bounded convex subset of a
topological vector space has an approximate fixed point sequence. Moreover, it
is constructed a affine sequentially continuous map from a compact convex set
into itself without fixed points.
