Zeros of mean-periodic functions (Q1819307)

Let \(\omega >0\) be the period of a periodic in mean continuous function F and let E be a set with the property \[ [0,\omega)\subset \{x\in {\mathbb{R}};\quad x=y-[y/\omega]\omega \text{ for some } y\in E\}. \] Than the following implication \(f|_ E=0\Rightarrow f=0\) is correct. The author proves that this implication is non correct for continuous \(\omega\)-periodic in mean functions even if \(E=[a,b]\cap [c,d]\).