Suppose you have a continuously differentiable function f: R -> R with |f'(x)| <= c < 1 for all x in R. Define a second function [tex]F: R^2 \rightarrow R^2[/tex] by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]F(x,y) = (x + f(y), y + f(x)).[/tex]

F is supposed to be onto. Why is that so?

Intuitively, I would say that F islocallyonto by the Inverse Function Theorem (since det(DF) = 1 - c^2 > 0). And globally the identity portion of F dominates over the little perturbation from f, so I would expect the function to be onto. But that's far away from a proof. Any suggestions?

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# Why is this function onto?

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