If the glide slope fails, you most certainly do not need to go missed. In many cases it's the best thing to do, but there are times and places where it's completely acceptable to continue the approach.
Last year I experienced a glide slope failure in the soup while shooting an approach. I had prebriefed the localizer approach, and on elected to continue localizer only. It was a parallel approach, and shortly the localizer flagged. After querying ATC, we were offered the parallel, and we took it. The glide slope flagged. We queried ATC, and were advised that "everything is going down due to the storm."
We continued localizer only, and it failed about the time we got the runway. We landed (and our T/R's failed to arm...it was one of those days).
Aircraft behind us ended up having to divert. The weather wasn't good, and there were no close alternates. When we filed, an alternate wasn't even necessary, but things had gone from bad to worse.
Preparation is essential if one is to modify the approach while performing the approach. Certainly being familiar with the proceedures is important. I say proceedures because the loc only and the full gs approach may vary considerably. One must be fully oriented to the approach and aware of position and capability to finish the approach. The loc only had better not be overly complicated, as some tend to be. There are many factors which may affect the operation (not the least of which are your own capabilities, the aircraft, the proceedure(s), the facilities, etc).
If performing a full ILS and experiencing a failure at or below loc only minimums, then no effort should be made to continue the approach unless the field is in sight. Simply execute the missed.
Not all ILS approaches have a marker to identify the missed approach point. Timing may be necessary for identification, as well as specific airspeed control to validate the timing effort.
Ali, yes, it's both safe and legal if done properly. However, going missed and executing the full loc only proceedure is also safe, and legal. If you're already burned up your reserves and are at your alternate now...perhaps it's not so safe to go around and try it again. Use best judgement based on all the factors, and it's just another tool in your reportoire.