Suppose $H$ is the reproducing kernel Hilbert space on a space $X$ with reproducing kernel $K$. If, say, $K - c\delta$ is a positive definite kernel for some $c>0$ (where $\delta(x,y) = \delta_{x,y}$) then $H$ contains the constant functions on $X$. But this condition is not necessarily easy to verify.

Are there other known sufficient conditions that imply that a reproducing kernel Hilbert space on a space $X$ contains the constant functions? In the situation I'm interested in $X$ is a metric space, so I don't have access to Fourier analysis and things like Bochner's theorem.