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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.
If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous. [5] Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous. [6] If f ...
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 f {\displaystyle f} is a linear operator between Banach spaces with closed graph, or if f {\displaystyle f} is a map with closed graph between compact ...
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to . The converse does not hold, since the function f : R → R , x ↦ x 2 {\displaystyle f:R\rightarrow R,x\mapsto x^{2}} is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous.
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue). In calculus , the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval [ a , b ] {\displaystyle [a,b]} , then f {\displaystyle f} must attain a maximum and a ...
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...
Alternatively, if s is an increasing function then s is continuous if s: γ → range(s) is a continuous function when the sets are each equipped with the order topology. These continuous functions are often used in cofinalities and cardinal numbers. A normal function is a function that is both continuous and strictly increasing.