There is at most one continuous invariant mean

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping $\mathbf{M}\text{\hspace{0.17em}⁣}:I^p\to I^p$ (where $I$ is an interval and $p \in $\mathbb{N}$$),thefunctionalequation$K \circ $\mathbf{M}$=K$ has at most one solution in the family of continuous means $K\text{\hspace{0.17em}⁣}:I^p\to I$. Some general approach to the latter equation is also given.