Last half-open and terribly rude (6)
I believe the answer is:
endure
'last' is the definition.
(I know that last can be written as endure)
'half-open and terribly rude' is the wordplay.
'half' suggests halving the letters.
'and' is a charade indicator (letters next to each other).
'terribly' is an anagram indicator.
'open' with half the letters taken is 'en'.
'rude' with letters rearranged gives 'dure'.
'en'+'dure'='ENDURE'
(Other definitions for endure that I've seen before include "'Last, bear (6)'" , "Last or undergo" , "Stand" , "'Last, prevail (6)'" , "Tolerate, put up with" .)