  Let $C(K)$ be the space of all real or complex valued continuous functions on
a compact Hausdorff space $K$. We are interested in the following property of
$K$: for any real valued $f \in C(K)$ the double commutant of the corresponding
multiplication operator $F$ coincides with the norm closed algebra generated by
$F$ and $I$. In this case we say that $K \in \mathcal{DCP}$. It was proved in
\cite{Ki} that any locally connected metrizable continuum is in
$\mathcal{DCP}$. In this paper we indicate a class of arc connected but not
locally connected continua that are in $\mathcal{DCP}$. We also construct an
example of a continuum that is not arc connected but is in $\mathcal{DCP}$.
