  The primary goal of this paper is to complete the theory of metric
Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007),
p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here
for the first time we obtain fully explicit bounds for the number of rational
points near planar curves. Further, introducing a perturbational approach we
bring the smoothness condition imposed on the curves down to $C^1$ (lowest
possible). This way we broaden the notion of non-degeneracy in a natural
direction and introduce a new topologically complete class of planar curves to
the theory of Diophantine approximation. In summary, our findings improve and
complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and
extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of
Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of
non-degeneracy.
