Consider a Mercer Kernel $K\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$, $\mathcal{X}$ being a compact subset of $\mathbb{R}^m$, and its (unique) associated Reproducing Kernel HIlbert Space $H_K$. Then $H_k\subset \mathcal{C}^0(\mathcal{X})$, where $\mathcal{C}^0(\mathcal{X})$ is the space of continuous function on $\mathcal{X}$, i.e., there exist some functions of $\mathcal{C}^0(\mathcal{X})$ which are not in $H_K$.
Could you give me an explanation of the latter statement? And/or could you give me an example of functions in $\mathcal{C}^0(\mathcal{X})$ which do not belong to $H_K$?
Thank you.