Let $f\in L^\infty(\mathbb{R})\cap C(\mathbb{R})$, that is $f$ is continuous and bounded on $\mathbb{R}$. Let $S\_r$ denote the shift by $r\in \mathbb{R}$: $S\_r f=f(\cdot-r)$.


Suppose $S\_{r} f $ converges to $ f$ as $r\rightarrow 0$ in the weak dual topology $\sigma(L^\infty, L^1)$, for a that is, for each $\varphi \in L^1(\mathbb{R})$, we have
$$
\int\_{\mathbb{R}} (S\_r f)\varphi dx \rightarrow \int\_{\mathbb{R}} f \varphi \textrm{ as } r\rightarrow 0.
$$
**Question:** Is there a subnet of $(S\_r f)\_{r>0}$ on which the convergence is in $L^\infty(\mathbb{R})$?



Convergence in weak dual topology $\sigma(L^\infty, L^1)$
It isn't research level, but $f(t) = \sin(e^{t^2})$ is a counterexample. (The uniform distance between $f$ and any shift of $f$ is $1$.)


