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Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above. Let : be a function defined on a subset of the set of real numbers. This subset is the domain of f. Some possible choices include
Continuous function; Absolutely continuous function; Absolute continuity of a measure with respect to another measure; Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous.
A stronger statement is that energy is locally conserved: energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.
If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology.
This is another way to say that the most chaotic behavior of f is concentrated on the support of the equilibrium measure. Finally, one can say more about the dynamics of f on the support of the equilibrium measure: f is ergodic and, more strongly, mixing with respect to that measure, by Fornaess and Sibony. [11]
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