Are there non-continuous kernels?

Question

In Mercer's theorem context, the kernel has to be a continuous function. But if you look at the RKHS theory, the kernel does not have to be continuous.

Is it possible to find a symmetric, positive semidefinite kernel which is non-continuous?

Thanks

Answer 1

Yes there are non-continuous positive semi-definite kernels. One actually useful example is the kernel \begin{equation} K(x,y)= \begin{cases} 1, & \text{ if } x=y \\ 0, & \text{ if }x\neq y. \end{cases} \end{equation} This is positive definite as it is the covariance function of the Gaussian white noise process, i.e. independent unit normal. Adding this kernel to a continuous one is the standard way to include noise/uncertainty in Gaussian process regression.