  In this paper we continue the study initiated by Gurariy and Quarta in 2004
on the existence of linear spaces formed, up to the null vector, by continuous
functions that attain the maximum only at one point. Inserting a topological
flavor to the subject, we prove that results already known for functions
defined on certain subsets of R are actually true for functions on quite
general topological spaces. In the line of the original results of Gurariy and
Quarta, we prove that, depending on the desired dimension, such subspaces may
exist or not.
