A bijection is a one-to-one correspondence in set theory - a function which is both a surjection and an injection.

A bijection is a one-to-one correspondence in set theory - a function which is both a surjection and an injection.

Inverse of a function exists only if it is a Bijection. Bijection=Injection(one to one)+surjection (onto) function.

Too long to explain so just go here http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

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.