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Romania District Olympiad 2003 - Grade XI

Source:

March 18, 2011
functionreal analysisreal analysis unsolved

Problem Statement

Let α>1\alpha>1 and f:[1α,α]→[1α,α]f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right], a bijective function. If f−1(x)=1f(x), ∀x∈[1α,α]f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right], prove that:
a)ff has at least one point of discontinuity; b)if ff is continuous in 11, then ff has an infinity points of discontinuity; c)there is a function ff which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity.
Radu Mortici
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