Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable

$$ X_\alpha = \lambda( \{t \; : \; Z(t) > \alpha \}) $$

where $\lambda$ is Lebesgue measure. $X_\alpha$ is a random variable taking values in $[0,1]$. Another thing to note is that for many Gaussian processes the distribution of $X_\alpha$ will have mass at 0 and 1. My question is: under what conditions on $C$ is $$ P( X_\alpha = x ) = 0, \mbox{ for all } x \in (0,1)? $$ Clearly this holds for many well known Gaussian processes, e.g. Brownian motion, and does not hold for many others (for example, suppose $Z(t) = N\sin(2\pi t)$, where $N\sim \mathcal{N}(0,1)$). I hoped there would be some result in the literature giving guidance on this, but I cannot find anything. This is a crosspost from the Math Stack Exchange, but after several months and bounties no-one there was able to provide any insight.