Uniformly convergence of delta sequences

Question

Let $(f_n) $ be a sequence of continuous functions $f_n: \mathbb R \rightarrow \mathbb R$ such that $$ \lim_{n\rightarrow \infty}\int_{\mathbb R} f_n(x) \phi(x) dx=\phi(0) $$ for each continuous compactly supported function $\phi :\mathbb R \rightarrow \mathbb R$. Is it then $(f_n)$ uniformly convergent to $0$ on each $[a,b] \subset \mathbb R\setminus \{0\}$ ?