Geodesic
Almost-Linear Segments of Graphs of Functions
Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 679-686.

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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.
Mots-clés : rectifiable curve
Keywords: graph of a function, discrete geometry. 
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     author = {A. M. Zubkov and O. P. Orlov},
     title = {Almost-Linear {Segments} of {Graphs} of {Functions}},
     journal = {Matemati\v{c}eskie zametki},
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A. M. Zubkov; O. P. Orlov. Almost-Linear Segments of Graphs of Functions. Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 679-686. http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a3/

[2] H. Sagan, Space-Filling Curves, Springer-Verlag, New York, 1994 | MR | Zbl

[3] V. Shmidt, Diofantovy priblizheniya, Mir, M., 1983 | MR