An empty function, is a function whose domain is the empty set. For each set A, there is exactly one such empty function fA: ∅ → A.
This includes a constant formula c; but the range of the empty function is not {c} but instead is the empty set.
Most authors will not care, when defining the term "constant function" precisely, whether or not the empty function qualifies, and will use whatever definition is most convenient.
Sometimes, however, it is best not to consider the empty function to be constant, and a definition that makes reference to the range is preferable in those situations.
(This is much along the same lines of not considering an empty topological space to be connected, or not considering the trivial group to be simple.)