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 Mat. Zametki, 2019, Volume 106, Issue 5, Pages 679–686 (Mi mz12416)

Almost-Linear Segments of Graphs of Functions

A. M. Zubkova, O. P. Orlovb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Lomonosov Moscow State University

Abstract: 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.

Keywords: rectifiable curve, graph of a function, discrete geometry.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm12416

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English version:
Mathematical Notes, 2019, 106:5, 720–726

Bibliographic databases:

UDC: 514.753.22+511.9

Citation: A. M. Zubkov, O. P. Orlov, “Almost-Linear Segments of Graphs of Functions”, Mat. Zametki, 106:5 (2019), 679–686; Math. Notes, 106:5 (2019), 720–726

Citation in format AMSBIB
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