A monotonic, or one-to-one function.

that's not really a question?

No, they can only be jump continuous.

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.

Here are some:

odd, even; periodic, aperiodic;

algebraic, rational, trigonometric, exponential, logarithmic, inverse;

monotonic, monotonic increasing, monotonic decreasing,

real, complex;

discontinuous, discrete, continuous, differentiable;

circular, hyperbolic;

invertible.