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No.
For example, y = 7 is monotonic. It may be a degenerate case, but that does not disallow it. It is not a bijection unless the domain and range are sets with cardinality 1.
Even a function that is strictly monotonic need not be a bijection. For example, y = sqrt(x) is strictly monotonic [increasing] for all non-negative x. But it is not a bijection from the set of real numbers to the set of real numbers because it is not defined for negative x.
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Is every bijection a strictly monotonic function?
No. For example, consider the discontinuous bijection that increases linearly from [0,0] to [1,1], decreases linearly from (1,2) to (2,1), increases linearly from [2,2] to [3,3], decreases linearly from (3,4) to (4,3), etc.
What is a bijection?
A bijection is a one-to-one correspondence in set theory - a function which is both a surjection and an injection.
When does a relation be a function?
A function is a relation whose mapping is a bijection.
What is the difference between relations and functions?
A function is a relation whose mapping is a bijection.
Is a strictly monotonic function absolutely continuous?
No, they can only be jump continuous.
