Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, $\{A_n\}_n$ a sequence of self-adjoint operators in $\mathcal{B}\left(\mathcal{H}\right)$ (the bounded linear operators on $\mathcal{H}$), which converges in norm to some self-adjoint $B\in\mathcal{B}\left(\mathcal{H}\right)$.
We know (it's easy to prove) that if $f:\mathbb{R}\to\mathbb{R}$ is a polynomial then $\{f\left(A_n\right)\}_n$ converges in norm to $f\left(B\right)$.
We also know (for example, from Reed & Simon theorem VIII.20) that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function vanishing at infinity then $\{f\left(A_n\right)\}_n$ converges in norm to $f\left(B\right)$. This theorem uses the Stone-Weierstrass theorem, which needs a compact domain.
My question is: what about continuous functions $f:\mathbb{R}\to\mathbb{R}$ which do not vanish at infinity? Is there a counter-example to that statement, or is it true? If yes, how to prove it? We know that if $f$ is merely bounded and $A_n$'s and $B$ are allowed to be unbounded then the convergence is merely strong (the same theorem from Reed & Simon), but perhaps with the extra assumption of boundedness of the original operators things could be strengthened.