  The sheaf-function correspondence identifies the group of constructible
functions on a real analytic manifold $M$ with the Grothendieck group of
constructible sheaves on $M$. When $M$ is a finite dimensional real vector
space, Kashiwara-Schapira have recently introduced the convolution distance
between sheaves of $k$-vector spaces on $M$. In this paper, we characterize
distances on the group of constructible functions on a real finite dimensional
vector space that can be controlled by the convolution distance through the
sheaf-function correspondence. Our main result asserts that such distances are
almost trivial: they vanish as soon as two constructible functions have the
same Euler integral. We formulate consequences of our result for Topological
Data Analysis: there cannot exists non-trivial additive invariants of
persistence modules that are continuous for the interleaving distance.
