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Generalizations of a “folk-theorem” on simple functional equations in a single variable. (English) Zbl 0725.39003
The “folk theorem” the authors start with is the following “Theorem 1. The general solution of (1) $$f(2x)=2f(x)$$ on $${\mathbb{R}}_+=]0,\infty [$$ is given by $$f(x)=xp(\ln x/\ln 2)$$, where p is an arbitrary periodic function with period 1. The solution is continuous, n-times differentiable, $$C^ n$$ (n$$\leq \infty)$$ or analytic on $${\mathbb{R}}_+$$ if p has the respective properties on $${\mathbb{R}}.$$
If f is defined at $$x=0$$ and (1) is satisfied then $$f(0)=0$$. If f is defined on [0,a[ and bounded on $$[2^{-m},2^{-m+1}[$$ for one m then f is continuous at 0. However, every solution, differentiable from the right at 0 (and defined on a right neighbourhood of 0, where it satisfies (1)) is of the form $$f(x)=cx.''$$
In the rest of the paper the authors (cf. the abstract) “... present similar and further results concerning general, $$C^ n$$ (n$$\leq \infty)$$, analytic, locally monotonic or $$\gamma$$-th order convex solutions of the somewhat more general equation $$f(kx)=k^{\gamma}f(x)$$ (k$$\neq 1$$ a positive, $$\gamma$$ a real constant), which seems to be of importance in meteorology...”.

##### MSC:
 39B22 Functional equations for real functions
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##### References:
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