  We show that, for a (not necessarily continuous) weakly contractive mean-type
mapping $\mathbf{M} \colon I^p\to I^p$ (where $I$ is an interval and $p \in
\mathbb{N}$), the functional equation $K \circ \mathbf{M}=K$ has at most one
solution in the family of continuous means $K \colon I^p \to I$.
  Some general approach to the latter equation is also given.
