# Nearly eventually almost periodic functions

Call a function $$f: [0, \infty) \to \mathbb R$$ nearly eventually almost periodic with period $$p > 0$$ if for a.e. $$x \in [0, p)$$, the sequence $${f(x + np)}_{n \in \mathbb N}$$ converges.

Suppose $$f: [0, \infty) \to \mathbb R$$ is continuous and eventually almost periodic of periods $$1$$ and $$a$$, where $$a$$ is irrational and $$0 < a < 1$$. Define $$F: [0, 1) \to \mathbb R$$ by $$F(x) := \lim_{n \to \infty} f(x + n)$$. Is $$F$$ necessarily constant a.e.? That is, is $$F$$ equal a.e. to a constant function?

• You already posted a very similar question. The very least you should do is link to it and point out the difference, and perhaps comment thereon (I imagine you're changing the question because you're unsatisfied with the answer you got, but you should state this). – Gro-Tsen Mar 9 at 11:55