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An incompatibility theorem. (English) Zbl 0544.26004

In this note the following incompatibility theorem is presented.
Theorem. If f and g are functions on \(R\times R\) into R, the reals, such that: a) \(f_ x\), \(f_ y\), \(g_ x\), and \(g_ y\) (partial derivatives) exist and are continuous on an open neighbourhood of the origin; \(b)\quad f_ x(0,0)\neq 0, f_ y(0,0)\neq 0, g_ x(0,0)\neq 0, g_ y(0,0)\neq 0; c)\quad f(0,0)=g(0,0)=0;\) and d) the numbers \(f_ x(0,0)/f_ y(0,0)\) and \(g_ x(0,0)/g_ y(0,0)\) have opposite signs; then there exist sets A, B; \(A,B\subset R;\) such that \(f(A\times B)\) contains an interval, but \(g(A\times B)\) does not. This result is related to various theorems of M. Kuczma, M. E. Kuczma, W. Sander, H. Steinhaus and the current author.

MSC:

26B35 Special properties of functions of several variables, Hölder conditions, etc.
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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