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?