Are there any functions that satisfy these properties?

Question

You're given a function $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ that satisfy these properties $$ f(x +1)=f(x), f(n + \dfrac{1}{2})=1, f(n)=0 \ \forall x \in \mathbb{R}, n \in \mathbb{N}$$ Are there any functions $m,n$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfy

$$ m(x) \neq 0 \ \ \text{with all $x$ satisfy} \ \ f(x)=m(x)(n(x) + 1) $$

Answer 1

Note that $n(x) = f(x) - 1$, $m(x) = 1$ satisfy the conditions of the problem. Here is a specific example of $f(x), n(x), m(x)$that work for $x\in \mathbb{R}$:

$$f(x) = \frac{-\cos(2\pi x) + 1}{2}$$

$$n(x) = \frac{-\cos(2\pi x) - 1}{2}$$

$$m(x) = 1$$