TRACON Controller described a low altitude event when a non-Mode C IFR departure apparently was not issued a climb to the appropriate altitude; the reporter noting position relief distractions and malfunctioning Mode C as a causal factors.

Narrative

King Air departed with a non-operational Mode-C. I had them report their altitude and try and re-cycle their transponder. I then also turned them to a heading of 340 to get them on course. I thought I climbed them to 14;000 as is our [standard] hand off with the Center. In the meantime; a controller change was in progress and I said I would coordinate with the Center that the aircraft Mode-C was inoperative. As I was doing so; the King Air reported a terrain warning to which the other controller climbed them immediately and asked if they had been assigned 14;000. They responded that they had not been. Contributing to the problem was I asked them what altitude they were at so I could issue a point out to another sector and I believed they said they were leaving 4;000; but they must have said they were level at 4;000. I think I would have checked his altitude on my own prior to the altitude warning and realized I forgot to climb them if a controller change was not in progress; so that was a cause. Also not being able to see the altitude was obviously the main problem; but I still should have made sure they were climbing. Always double check the altitude of someone with an invalid Mode-C even if you think you climbed them.