About the density of a subqpace of $L^2$?

Question

We all know that $C_c^{\infty}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$. Now, is the space $\{f\in C_c^{\infty}(\mathbb{R}):f(a)=0\}$ (where $a\in\mathbb{R}$) still dense in $L^2(\mathbb{R})$?

Thanks a lot!

Math.