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the sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition.
In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. In advanced theory, the applications of this functional calculus are so natural that they are often not even mentioned.
For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone. In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.
Let C 0 (X, R) be the space of real-valued continuous functions on X that vanish at infinity; that is, a continuous function f is in C 0 (X, R) if, for every ε > 0, there exists a compact set K ⊂ X such that | f | < ε on X \ K. Again, C 0 (X, R) is a Banach algebra with the supremum norm.
The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.
The restriction of a continuous function is continuous. [3] [4] Applications. Inverse functions. For a function to have an inverse ... In relational algebra, ...