# Convergence in weak dual topology $\sigma(L^\infty, L^1)$

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

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 $$1$$.)