Wheelybin's reply is interesting. Many approach controllers are clearly happy to provide FIS or RIS to traffic outside their zone or CTA, but it is certainly not universal.
My experience is that Luton have not provided FIS to traffic (unless it wishes to transit) since they stopped providing a LARS service last year. Mostly a request for FIS is met with an instruction to call London Information. It would seem that they do not wish to know about traffic which is quite legally flying close to the lateral or vertical boundary of their airspace. Does this cause problems in acheiving IFR seperation? Is the radar controller required to achieve 5m lateral seperation (or vertical seperation, if unverified mode C squawk is seen) from unknown traffic which is outside his CTR/CTA/TMA? This would seem to dramatically reduce the useful size of the controlled airspace.
However I have also found Luton helpful, on IMC days, in providing a radar service outside their CTA -- thanks for this, it is really appreciated, even on a 'controller workload permitting' basis.
Birmingham have also issued a Notam which explains that they may limit their radar service to aircraft in their CTA, and may also not allow transits to traffic not intending to land at Birmingham. Until recently, I had never been refused a transit, but last week (Thursday pm), I was told to 'remain clear.' I was offered FIS, (without squawk, suggesting that seperation of their controlled traffic from me was not their intention.
Essex, on the other hand, despite their reputation for lack of interest in GA traffic, have often offered me a service outside CAS in IMC, and have often offered a transit of the Stansted CTA. Again, much appreciated.
So why are there variations?
Regards.