Let $X$ be a Banach space with norm $\|\cdot\|$. If $g:\mathbb{R}\to\mathbb{R}$ is defined by: \begin{equation} g(s)=\begin{cases} \exp[-s^{-2}] & s>0 \\ 0 & s\leq 0 \end{cases} \end{equation} then, \begin{equation} b(x)=\frac{g(2-\|x\|)}{g(\|x\|-1)+g(2-\|x\|)} \end{equation} is a (norm-norm) continuous bump function such that $b(0)=1$ and $b(x)=0$ for all $\|x\|\geq 2$. Does there exist an analogous construction that yields a weak-norm continuous (i.e. continuous with respect to the norm topology on $\mathbb{R}$ and the weak topology on $X$) bump function?
More generally, is there such a thing as a weak-norm continuous bump function? I have not been able to find a reference anywhere for such an object. Thank you in advance for any insights!