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Romania National Olympiad 2009 - Grade XI

Source:

April 10, 2011
functionlimitreal analysisreal analysis unsolved

Problem Statement

Let f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} a continuous function such that for any x∈Rx\in \mathbb{R}, the limit lim⁡h→0∣f(x+h)−f(x)h∣\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right| exists and it is finite. Prove that in any real point, ff is differentiable or it has finite one-side derivates, of the same modul, but different signs.
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