  We study the problem of the existence of increasing and continuous solutions
$\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$ of the
functional equation \begin{equation*}
\varphi(x)=\sum_{n=0}^{N}\varphi(f_n(x))-\sum_{n=1}^{N}\varphi(f_n(0)),
\end{equation*} where $N\in\mathbb N$ and $f_0,\ldots,f_N\colon[0,1]\to[0,1]$
are strictly increasing contractions satisfying the following condition
$0=f_0(0)<f_0(1)=f_1(0)<\cdots<f_{N-1}(1)=f_N(0)<f_N(1)=1$. In particular, we
give an answer to the problem posed in the article Remark on BV-solutions of a
functional equation connected with invariant measures by Janusz Matkowski
concerning a very special case of that equation.
