Proceedings of the 2021 Undergraduate Mathematics Day
This paper concerns fixed points of functions whose graphs lie on or below the line y = x. Using the Monotone Convergence Theorem, we show that positive fixed points of such functions are “attracting on the right” so long as we include a couple of further assumptions about these functions near their fixed points. As an illustrative example, we confirm that this is the case for the function y = x sin x; the positive fixed points of this function “attract on the right” and “repel on the left.” Further, we generalize by showing that differentiability is in fact not needed to conclude that a fixed point is attracting on the right. Continuing in this direction, we identify a class of discontinuous functions whose fixed points are attracting on the right.
Fixed points of real valued functions, attracting fixed point, repelling fixed point
Fryling, Grace and Rouse, Harrison, "Fixed Points of Functions below the Line y = x" (2022). Undergraduate Mathematics Day: Proceedings and Other Materials. 44.