Continue to look up (4)
I believe the answer is:
keep
'continue' is the definition.
(keeping is a kind of continuing)
'look up' is the wordplay.
'look' becomes 'peek' (peek is a kind of look).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'peek' reversed gives 'KEEP'.
'to' is the link.
(Other definitions for keep that I've seen before include "Retain, maintain" , "Retain or care for" , "Retain; strong tower" , "Act as custodian of" , "Central tower of castle" .)