In the literature, a tempered distribution $f \in \mathcal{S}'(\mathbb{R})$ is called almost-periodic if the convolution $f * \phi$ is almost-periodic in the classical sense for every test function $\phi \in C_c^{\infty}(\mathbb{R})$.
I am unsure if, or why this definition is consistent with the classical definition for continuous functions (i.e. relatively dense set of $\varepsilon$-translates).
In particular, is it possible for there to be a continuous function $f$ such that the convolution $f*\phi$ is almost periodic for every $\phi$, but $f$ is not almost-periodic in the classical sense? If $f$ is uniformly continuous I am happy there is no problem, but I cannot find a reference for the general case.