  We investigate a compact star in the general $F(R)$ gravity. Developing a
novel formulation in the spherically symmetric and static space-time with the
matter, we confirm that an arbitrary relation between the mass $M$ and the
radius $R_s$ of the compact star can be realized by adjusting the functional
form of $F(R)$. Such a degeneracy with a choice of the equation of state (EOS)
suggests that only mass-radius relation is insufficient to constrain the $F(R)$
gravity. Furthermore, by solving the differential equation for $\left.
\frac{dF(R)}{dR}\right|_{R=R(r)}$ near and inside the surface of the compact
star with the polytropic EOS, the boundary condition demands a weak curvature
correction to the Einstein gravity could be non-integer power of the scalar
curvature, which gives a stringent constraint on the functional form of
$F\left(R\right)$. This consequence follows that the equation of motion in
$F(R)$ gravity includes the fourth-order derivative of metric, and thus, it is
applicable to general $F(R)$ models.
