The most common and most likely answer to this crossword clue is the 13 letter word UNINTERRUPTED
I've seen this crossword clue in The Guardian
U
N
I
N
T
E
R
R
U
P
T
E
D
More clues leading to the result UNINTERRUPTED
Buddy explains!
The straight part
We know that "Constant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case constant = uninterrupted.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does UNINTERRUPTED mean?
UNINTERRUPTED adjective- Continuing with no interruption