Is the restriction of a test function still a test function?

Question

Let $D$ be an open set of $\mathbb{R}^{m}$ with $m\geq1$. Suppose $\phi\in C_{c}^{\infty}(D)$ and $F$ is a closed set included to $D$. Can we say that the restriction of $\phi$ to $D\setminus F$ is in $ C_{c}^{\infty}(D\setminus F) $?

Answer 1

No.

For example, we can take $m=1$ and $D=(-2,2)$, $F=[-1,1]$. Take $\phi\in C^\infty_c(D)$ to be the standard bump function such that $\phi=1$ on $[-1,1]$ and $\phi=0$ outside $(-1.5,1.5)$. Then $\phi$ is not compactly supported on $D\backslash F$.