Almost-Linear Segments of Graphs of Functions

Abstract

Let f: ℝ → ℝ be a function whose graph {(x, f(x))}_x∈ℝ in ℝ² is a rectifiable curve. It is proved that, for all L < ∞ and ɛ > 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 ≤ x ≤ b, to the segment AB do not exceed ε|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 < ∞, there exists a straight line containing at least r points of this sequence.