Hi again

Can you get the keyless entry function turned off on a 2013 RRS?

Concerned if the keys close to the vehicle someone will get in!

No they won't, I was concerned about this so tried different ways of getting into the vehicle and found the only way to get in is having the key on you and touching the door handle. Even if you are near the car your passanger cannot open the door till you touch the door handle.
Welcome to the site by the way.

Think you will find that if you are more than a couple of feet away it won't let you in so it shouldn't be a problem. MY16 AUTOBIOGRAPHY Indus silver with some toys

2012 3.0 SDV6 Luxury gone, but a great motor

If I am close enough to touch the car SWMBO can open the door but if I step back she can't - basically if someone is stood between you and the car they could open the door but then they wouldn't be able to get away or do much harm so not an issue - using remote buttons actually has more risk but who wants to go back to putting key in lock, etc  Wanted a Series 2 LR since childhood but previously owned MY16 Disco Sport HSE TD4 Auto, MY13 RR Sport Black Edition TDV6 Auto, MY10 RR Sport HSE TDV6 Auto, 2007 Freelander 1 Freestyle TD4 Soft Top, 2009 Freelander 2 GS TD4 Auto, 2007 Freelander 2 GS TD4, 2004 Disco 2 Metropolis Auto, 2002 Disco 2 GS, 2000 Freelander 1 SE TD4 SW

Still hoping for a S2 one day!