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Almost-linear segments of graphs of functions. (English. Russian original) Zbl 07179655
Math. Notes 106, No. 5, 720-726 (2019); translation from Mat. Zametki 106, No. 5, 679-686 (2019).
Summary: Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ 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 $$( x, f (x))$$, $$a \leq x \leq b$$, 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.
##### MSC:
 26 Real functions
##### Keywords:
rectifiable curve; graph of function; discrete geometry
Full Text:
##### References:
 [1] Sagan, H., Space-Filling Curves (1994), New York: Springer-Verlag, New York · Zbl 0806.01019 [2] Schmidt, W., Diophantine Approximation (1980), Heidelberg: Springer-Verlag, Heidelberg · Zbl 0421.10019
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