This is part of Exercise 1.15 from Brownian Motion, Martingales, and Stochastic Calculus by Jean-François Le Gall:
Let $(X_t)_{t \in [0,1]}$ be a centered Gaussian process. We assume that the mapping $(t, \omega) \mapsto X_t(\omega)$ from $[0,1]\times \Omega$ into $\mathbb{R}$ is measurable. We denote the covariance function of $X$ by $K$.
Show that the mapping $t \mapsto X_t$ from $[0,1]$ into $L^2(\Omega)$ is continuous if and only if $K$ is continuous on $[0,1]^2$. ...
It doesn't say in what sense the mapping is continuous so I assume that it's continuous in the almost surely sense.
I don't know how to approach this. An initial approach was to argue that we want to show $P(\forall t \in [0,1] \exists \{t_n\}_{n \geq 1} \text{ s.t. } t_n \to t \text{ and } X_{t_n} \nrightarrow X_t) = 0$. By a subset argument you could say it's sufficient to show this holds for a fixed $t$, and by another notion of convergence, if we define $B_n = \overline{B}(t, \frac{1}{n}) \cap [0,1]$, we could then try to show $P(\lim_{n \to \infty} \sup_{s \in B_n} |X_s - X_t| \neq 0) = 0$. By thinking of this as the limit being greater than zero i.o., we could maybe try to show that $P(\sup_{s \in B_n} |X_s - X_t|)$ is summable, and thus that this event does not happen infinitely often.
Then I get stuck.
I've tried reading papers and it looks like the general continuity properties of Gaussian processes are difficult to show. Supposedly this problem was used in exams, so maybe there is some trick for this space I'm missing?