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Take $X:=\mathbb{R}$ and $D:=\mathbb{R}\setminus\{0\}$. Consider any sequence of continuous functions $(f_n)_n$ that converges uniformly to $0$ on compact sets of $D$, but with $\langle \delta_0,f_n\rangle:=f_n(0)=1$, like e.g. $f_n(x):=(1-n|x|)_+$ .