Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, suppose the kernel is $K(x,y)$, since $K(x,\cdot) \in \mathcal{K} $, $K$ is continuous seperately in $x$ and $y$. But it need not be continuous in two variable.
Please someone give me an example that Reproducing kernel need not be continuous in two variables.
Let $1=a_0=a_1>\ldots >a_n>\ldots >0$, and let $e_n$ be the "triangle" function that vanishes outside $(a_{n+1},a_{n-1})$, equals $1$ at $a_n$, and interpolates linearly from $0$ to $1$ on $[a_{n+1},a_n]$ and from $1$ to $0$ on $[a_n,a_{n-1}]$.
Put $K(x,y)=\sum_1^\infty e_n(x)e_n(y)$. This is well defined, vanishes for $x=0$ (or $y=0$), is separately continuous on $[0,1]\times [0,1]$, is positive definite, and $K(x,x)$ oscillates between $1$ (at $a_n$) and $1/2$ (at $(a_{n+1}+a_n)/2)$ as $x\to 0$ (where $K(0,0)=0$).
The associated RKHS is the set of piecewise affine continuous functions $\sum_1^\infty c_ne_n$ with $\sum c_n^2<\infty$. The unit ball is not equicontinuous on $[0,1]$, i.e. it is not compact in $C([0,1])$, a necessary and sufficient condition for the continuity of $K$ on $[0,1]\times [0,1]$.
