Not a reason to give up — an inspiration (5)
I believe the answer is:
erato
'an inspiration' is the definition.
(one of the muses in Greek mythology)
'not a reason to give up' is the wordplay.
'to give' says the answer is hidden in the clue (I've seen 'gives' mean this).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'otare' is hidden within 'not a reason'.
'otare' written backwards gives 'ERATO'.
(Other definitions for erato that I've seen before include "poetic patroness" , "inspiration for poetry" , "Source of artistic inspiration" , "Who may have inspired poetry" , "Muse of lyric poetry and mime" .)