Minimal values

Let $f:𝕉\to 𝕉$ be a function (any function). Let's call

the set of points where $f$ has a (non-strict) local minimum. Let $V=f\left(M\right)$ 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).