  We find an asymptotic formula for the number of rational points near planar
curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a
sufficiently smooth function defined on the interval $[\eta,\xi]$, then the
number of rational points with denominator no larger than $Q$ that lie within a
$\delta$-neighborhood of the graph of $f$ is shown to be asymptotically
equivalent to $(\xi-\eta)\delta Q^2$.
