Function for mathematicians at home in the home counties (4)
I believe the answer is:
sine
'function for mathematicians' is the definition.
(mathematical function)
'home in the home counties' is the wordplay.
'home' becomes 'in' ('he's home' can mean 'he's in').
'in' is an insertion indicator.
'the home counties' becomes 'se' (Home Counties in the South East of England).
'in' inserted into 'se' is 'SINE'.
'at' is the link.
(Other definitions for sine that I've seen before include "Indefinitely, with no appointed date - . . . . die" , "Indispensable condition" , "Latin tag for indefinitely - . . . . die" , "Trig function" , "Opposite/hypoteneuse" .)