We consider the functional equation $f(xf(x))=\varphi (f(x))$
where $\varphi: J\rightarrow J$ is a given increasing
homeomorphism of an open interval $J\subset (0,\infty )$
and $f:(0,\infty )\rightarrow J$ is an unknown continuous
function. In a series of papers by P. Kahlig and J. Sm\'{\i}tal
it was proved that the range of any non-constant
solution is an interval whose end-points are fixed
under $\varphi$ and which contains in its interior no fixed point
except for $1$. They also provide a characterization of the class of
monotone solutions and proved a necessary and sufficient condition
for any solution to be monotone.
In the present paper we give a characterization of the class
of continuous solutions of this equation: We describe a method of
constructing solutions as pointwise limits of solutions
which are piecewise monotone on every compact subinterval. And we
show that any solution can be obtained in this way.
In particular, we show that if there exists a solution
which is not monotone then there is a continuous solution which is
monotone on no subinterval of a compact interval $I\subset (0,\infty )$.