I was told that in one dimension, if we consider a half open interval (a,b] then the space $C_{c}^{\infty}((a,b])$ is dense in $\tilde{H}^{1}((a,b)):=\{u\in H^{1}((a,b))\,|\,u(a)=0\}$. I am not sure how to show this.
I am familiar with density theorems for $H^{1}$ and $H^{1}_{0}$ but I am not sure how to progress when you have a mix between these two spaces. Can anyone offer a suggestion of how to consider this problem?
Since $C_{c}^{\infty}((a,b))\subseteq C_{c}^{\infty}((a,b])\subseteq C^{\infty}([a,b])$ then using the fact that $\overline{C_{c}^{\infty}((a,b))}^{H^{1}}=H^{1}_{0}((a,b))$ and $\overline{C^{\infty}([a,b])}^{H^{1}}=H^{1}((a,b))$ we have that $\overline{C_{c}^{\infty}((a,b])}^{H^{1}}=\tilde{H}^{1}((a,b))$.