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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 $L^\infty(\mathbb{R})$?

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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 $1$.)

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