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Romania District Olympiad 2004 - Grade XI

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April 10, 2011
functionreal analysisreal analysis unsolved

Problem Statement

Let f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} a function such that f(a+b2)∈{f(a),f(b)}, (∀)a,b∈Rf\left(\frac{a+b}{2}\right)\in \{f(a),f(b)\},\ (\forall)a,b\in \mathbb{R}.
a) Give an example of a non-constant function that satisfy the hypothesis.
b)If ff is continuous, prove that ff is constant.
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