If it was a GS failure then there is no reason why you can't continue as per the LLZ approach. If it is a loc failure then use a bit of common sense, level out or begin climbing towards MSA, maintain heading and ask ATC what they want you to do. When they clear you for the ILS they expect you to fly along the centerline until the middle marker then either land or follow the missed approach. I can assure you that they will NOT be pleased when you suddenly spear off to the left into their departures!