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On a local form of Lobachevski’s functional equation. (English) Zbl 0652.39009

For an open set \(S\subseteq {\mathbb{R}}\), f: \(S\to {\mathbb{R}}\) is “locally Lobachevskian” on S if, for all \(x\in S\), there exists a \(\delta (x)>0\) such that \((L)\quad f(x+h)f(x-h)=f(x)^ 2\) whenever \(0<h<\delta (x)\). The following is offered as main result. For f to be locally Lobachevskian on an open real interval, I, the following are necessary and sufficient. The set of zeros of f in I contains a half-neighbourhood of each of its elements; on each interval \(\tilde I\) contiguous to this set there exists an \(\tilde f:\) \(\tilde I\to {\mathbb{R}}\) satisfying (L) whenever \(x+h\), x-h\(\in \tilde I\). There exist a closed countable set \(M\subset \tilde I\) containing a symmetric neighbourhood around each of its elements, a sequence \(\{J_ n\}\) of intervals contiguous to M in \(\tilde I\) and a sequence \(\{a_ n\}\subset {\mathbb{R}}\) such that \(f| J_ n=a_ n\tilde f| J_ n\) \((n=1,2,...)\). Furthermore f is locally Lobachevskian on M.
Reviewer: J.Aczél

MSC:

39B99 Functional equations and inequalities
03E15 Descriptive set theory
03E20 Other classical set theory (including functions, relations, and set algebra)
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