Almost-Linear Segments of Graphs of Functions

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a function whose graph $\{(x, f(x))\}_{x \in \mathbb{R}}$ in $\mathbb{R}^2$ is a rectifiable curve. It is proved that, for all $L < \infty$ and $\varepsilon > 0$, there exist points $A = (a, f(a))$ and $B = (b, f(b))$ such that the distance between $A$ and $B$ is greater than $L$ and the distances from all points $(x, f(x))$, $a \le x \le b$, to the segment $AB$ do not exceed $\varepsilon |AB|$. An example of a plane rectifiable curve for which this statement is false is given. It is shown that, given a coordinate-wise nondecreasing sequence of integer points of the plane with bounded distances between adjacent points, for any $r < \infty$, there exists a straight line containing at least $r$ points of this sequence.