Continuous Replicative Function

Theorem
Let $f: \R \to \R$ be a real function.

Let $f$ be continuous on $\R$.

Let $f$ also be a replicative function.

Then $f$ is of the form:
 * $f \left({x}\right) = \left({x - \dfrac 1 2}\right) a$

where $a \in \R$.