Bombieri and Pila gave sharp estimates for the number of integer points $(m,n)$ on a given arc of a curve $y = F(x)$, enlarged by some size parameter $M$, for algebraic curves and for transcendental analytic curves. The transcendental case involves the maximum number of intersections of the given arc by algebraic curves of bounded degree. We obtain an analogous result for functions $F(x)$ of some class $C^k$ that satisfy certain differential inequalities that control the intersection number. We allow enlargement by different size parameters $M$ and $N$ in the $x$- and $y$-directions, and we also estimate integer points close to the curve, with
$$\left|n - NF ( {m\over M} )| \leq \delta,$$
for $\delta$ sufficiently small in terms of $M$ and $N$.
As an appendix we obtain a determinant mean value theorem which is a quantitative version of a linear independence theorem of Pólya.