  Entrywise functions preserving Loewner positivity have been studied by many
authors, most notably Schoenberg and Rudin. Following their work, it is known
that functions preserving positivity when applied entrywise to positive
semidefinite matrices of all dimensions are necessarily analytic with
nonnegative Taylor coefficients. When the dimension is fixed, it has been shown
by Vasudeva and Horn that such functions are automatically continuous and
sufficiently differentiable. A natural refinement of the aforementioned problem
consists of characterizing functions preserving positivity under rank
constraints. In this paper, we begin this study by characterizing entrywise
functions which preserve the cone of positive semidefinite real matrices of
rank $1$ with entries in a general interval. Classifying such functions is
intimately connected to the classical problem of solving Cauchy's functional
equations, which have non-measurable solutions. We demonstrate that under mild
local measurability assumptions, such functions are automatically smooth and
can be completely characterized. We then extend our results by classifying
functions preserving positivity on rank $1$ Hermitian complex matrices.
