The Oxford English Dictionary gives three definitions for the adjective "constant":
1. occurring continuously
2. remaining the same
3. faithful and dependable
While we typically use the word "constant" to mean something that occurs continuously, Shakespeare's usage typically means the second and third definitions.
An example from enotes is this dialogue from Julius Caesar:
"I am constant as the northern star,
Of whose true-fix'd and resting quality
There is no fellow in the firmament."
Julius Caesar Act III, Scene I.
With this statement, Caesar makes it clear that, despite three different warnings, he will not change his mind about going to the Senate on the Ides of March.