  Valadier and Hensgen proved independently that the restriction of functional
$\phi(x)=\int_{0}^{1}x(t)dt,\,\,x\in L^{\infty}([0,1])$ on the space of
continuous functions $C([0,1])$ admits a singular extension back to the whole
space $L^{\infty}([0,1]).$ Some general results in this direction for the
Banach lattices were obtained by Abramovich and Wickstead. In present note we
investigate analogous problem for variable exponent Lebesgue spaces, namely we
prove that if the space of continuous functions $C([0,1])$ is closed subspace
in $L^{p(\cdot)}([0,1]),$ then every bounded linear functional on $C([0,1])$ is
the restriction of a singular linear functional on $L^{p(\cdot)}([0,1])$.
