Wednesday, April 16, 2008

Minimal values

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function (any function). Let's call
$$ M := \{ x \in \mathbb{R} : \exists\epsilon_x>0: f(x) \leq f(y)\;whenever\;|y-x| < \epsilon_x\} $$
the set of points where $f$ has a (non-strict) local minimum. Let $V=f(M)$ be the set of the local minimal values. Can $V$ have positive Lebesgue measure? (note that $M$ surely can, just take as $f$ a constant function, and every point is a local minimum).

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