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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.
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 ...
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.
The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false. As an example, take the function f : [0, ∞) → [−1, 1] defined by f(x) = sin(1/x) for x > 0 and f(0) = 0.
Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. Minkowski's question mark function : Derivatives vanish on the rationals. Weierstrass function : is an example of continuous function that is nowhere differentiable
Linear functions + are the simplest examples of uniformly continuous functions. Any continuous function on the interval [ 0 , 1 ] {\displaystyle [0,1]} is also uniformly continuous, since [ 0 , 1 ] {\displaystyle [0,1]} is a compact set.
The function f(x) = √ x defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. However, it is uniformly continuous, [8] and both Hölder continuous of class C 0, α for α ≤ 1/2 and also absolutely continuous on [0, 1] (both of which imply the former).
There are examples of uniformly continuous functions that are not α –Hölder continuous for any α. For instance, the function defined on [0, 1/2] by f(0) = 0 and by f(x) = 1/log(x) otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however.
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