\(\varepsilon\)-regular functions (Q1381906)

The paper applies nonstandard analysis to study functions \(f: \overline U\to{\mathbb R}^n\) with \(U\subset{\mathbb R}^n\) open and convex. For \(\varepsilon>0\) infinitely small, \(f\) is called \(\varepsilon\)-regular if \(| f(x)-f(y)| /\varepsilon\) is along with \(| x-y| /\varepsilon\) neither infinitely small nor infinitely large. \(\varepsilon\)-regularity is shown to be equivalent to local (and in a certain case global) bilipschitzness. The implicit function theorem is given an equivalent nonstandard discrete, algorithmic form. Introducing \(\varepsilon\)-regular triangulations of \(U\) and \(\varepsilon\)-PL approximations of \(f\) and based on F. John's results it is given a sufficient nonstandard combinatorial condition for \(f\) to be locally bilipschitz. Global invertibility in a discrete form is also studied. Finally, a proof of a classical formulation of a quite general case of Liouville's theorem on characterization of conformal maps of \(({\mathbb R}^n,| \cdot| _i)\) for \(i\in\{1,2,\infty\}\) is reduced to the \(C^1\)-case if \(i=2\) and carried out completely if \(i=1\) or \(\infty\).