Can someone give an example of a discontinuous, positive semi-definite real function $f:\mathbb{R}\to\mathbb{R}$?
It is a well known fact that $f(t)= e^{-|t|}$ is a positive semi-definite real function. So perhaps the discontinuous function $g(t) = f(t)$ on $t\in\mathbb{R}\backslash \{0\}$ and $g(0)=2$ is also positive semi-definite, however I haven't been able to show it.