## On functions taking the same value on many pairs of points.(English)Zbl 1170.26006

Let $$I\subset\mathbb{R}$$ be an interval. A function $$f: I\to\mathbb{R}$$ is said to have the property $$(M_p)$$, where $$0< p< 1$$, $$p\neq{1\over 2}$$, if whenever $$x,y\in I$$ and $$f(x)\neq f(y)$$ then $$f(px+ (1- p)y)= f((1- p)x+ py)$$. The authors show that if $$f$$ is a measurable function with property $$(M_p)$$, then $$f$$ is constant a.e. Also, if “measurable” is replaced by “has a point of continuity in $$I$$”, then $$f$$ is constant apart from a countable set. The authors then apply this result to a certain functional equation.

### MSC:

 26B99 Functions of several variables 39B22 Functional equations for real functions

### Keywords:

functional equations; measurability
Full Text:

### References:

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