Let $\Omega, \Omega'$ be two open subsets of $\mathbb{R}^{n}$ such that $\Omega \subset \Omega'$. Let $f \in C_c^\infty(\Omega)$ and $\tilde{f}$ be the trivial extension of $f$ to $\Omega'$, that is, $\tilde{f}=f$ in $\Omega$ and $\tilde{f}=0$ in $\Omega'\setminus \Omega.$ Let $J:C_c^\infty(\Omega) \to C_c^\infty(\Omega')$ given by $J(f)=\tilde{f}$.
I already verified that $J$ is injective and continuous.
My question: Suppose that $\Omega \neq \Omega'$, how to prove that $J^{-1}:\operatorname{Im}(J) \subset C_c^\infty(\Omega') \to C_c^\infty(\Omega)$ is not continuous?
The spaces $C_c^\infty(\Omega)$ and $C_c^\infty(\Omega')$ are equipped with their respective $LF$ topologies.