  It is known that a positive, compactly supported function $f \in L^1(\mathbb
R)$ can tile by translations only if the translation set is a finite union of
periodic sets. We prove that this is not the case if $f$ is allowed to have
unbounded support. On the other hand we also show that if the translation set
has finite local complexity, then it must be periodic, even if the support of
$f$ is unbounded.
