  We study how an evanescence process affects the number of distinct sites
visited by a continuous time random walker in one dimension. We distinguish two
very different cases, namely, when evanescence can only occur concurrently with
a jump, and when evanescence can occur at any time. The first is characteristic
of trapping processes on a lattice, whereas the second is associated with
spontaneous death processes such as radioactive decay. In both of these
situations we consider three different forms of the waiting time distribution
between jumps, namely, exponential, long-tailed, and ultra-slow.
