Could we choose $\phi_m\in D(I-\{0\})$ to approximate $\phi\in D(I)$ ($\phi(0)=0$).

Question

Suppose $\phi\in D(-1,1)$ ($C_c^\infty$ space with standard topology) with $\phi(0)=0$. Does there exist $\phi_m\in D(-1,1)$, such that the support of $\phi_m$ is away from $0$, and $\phi_m\rightarrow\phi$ in $D(-1,1)$ as $m\rightarrow\infty$?

Answer 1

No, this is only possible if all derivatives of $\phi$ vanish at $0$, since $\phi_m \to \phi$ implies $0 = \phi^{(n)}(0)_m \to \phi^{(n)}(0)$ for all $n \ge 0$. However, an arbitrary $\phi \in D(-1,1)$ with $\phi(0) = 0$ might have $\phi'(0) \ne 0$.