When is $C_c^\infty(\mathbb R^d \setminus \{ 0 \})$ dense in $C_c^\infty(\mathbb R^d)$?

Let $b_n$ be a sequence of smooth functions $b_n: \mathbb{R} \to [0,1]$ such that $b_n(x) = 1$ for $|x|> { 2\over n}$ and $b_n(x) = 0$ for $|x|< {1 \over n}$.

If $f \in C_c^\infty(\mathbb R^d)$, then $b_n\cdot f \in C_c^\infty(\mathbb R^d \setminus \{0\})$ and for $p< \infty$ we have $\|f-b_n \cdot f\|_p \to 0$.

If $f \in C_c^\infty(\mathbb R^d)$ such that $f(0) = 1$, then we must have $\|f-g\|_\infty \ge 1$ for any $g \in C_c^\infty(\mathbb R^d \setminus \{0\})$.