Chap after party lapping up cheers — some function! (9)
I believe the answer is:
cotangent
'some function' is the definition.
(cotangent is a kind of function)
'chap after party lapping up' is the wordplay.
'chap' becomes 'Tan'.
'after party' becomes 'cogent' (I can't explain this - if you can you should believe this answer much more).
'lapping up' is an insertion indicator (to lap up is to drink or consume).
'tan' going inside 'cogent' is 'COTANGENT'.
'cheers' acts as a link.
This may not be correct. It may belong to another bit of the clue.
(Other definitions for cotangent that I've seen before include "proportion" , "In triangular relationship" , "Maths function" , "Mathematician's function" , "Mathematical ratio" .)