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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.
Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.
If one wants to extend the natural functional calculus for polynomials on the spectrum of an element of a Banach algebra to a functional calculus for continuous functions (()) on the spectrum, it seems obvious to approximate a continuous function by polynomials according to the Stone-Weierstrass theorem, to insert the element into these polynomials and to show that this sequence of elements ...
The space of complex-valued continuous functions on a compact Hausdorff space i.e. (,) is the canonical example of a unital commutative C*-algebra. The space X may be viewed as the space of pure states on , with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved ...
So, if the open mapping theorem holds for ; i.e., is an open mapping, then is continuous and then is continuous (as the composition of continuous maps). For example, the above argument applies if is a linear operator between Banach spaces with closed graph, or if is a map with closed graph between compact Hausdorff spaces.
the function f is n − 1 times continuously differentiable on the closed interval [a, b] and the n th derivative exists on the open interval (a, b), and; there are n intervals given by a 1 < b 1 ≤ a 2 < b 2 ≤ ⋯ ≤ a n < b n in [a, b] such that f (a k) = f (b k) for every k from 1 to n. Then there is a number c in (a, b) such that the n ...
The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if E is a Banach space with trivial centralizer and X and Y are compact, then every linear isometry of C(X; E) onto C(Y; E) is a strong Banach–Stone map.
For example, let X be the set of ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by "open intervals". The linear functional taking a continuous function to its value at Ω corresponds to the regular Borel measure with a point mass at Ω.
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