Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $W=\big\{W(t):t\in[0,T]\big\}$ be a Brownian motion on it. Suppose $\big\{\mathcal{F}(t):t\in[0,T]\big\}$ be the filtration generated by $W$, i.e., $\mathcal{F}(t)=\sigma\{W(u), u\leq t\}$, for all $t\in[0,T]$. Consider a random variable $Z$ and define the process $Z(t):=\mathbb{E}[Z|\mathcal{F}(t)], t\in[0,T]$. Is this necessarily a continuous time process?
A stochastic process $\big\{X(t):t\in[0,T]\big\}$ is said to be a continuous time process if for each fixed $w\in\Omega$(good enough for almost every $w$ in $\Omega)$, the function $t\mapsto X(t)(w)$ is continuous.