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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.
However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
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.
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 ...
A function: between two topological spaces is a homeomorphism if it has the following properties: . is a bijection (one-to-one and onto),; is continuous,; the inverse function is continuous (is an open mapping).
Definition: We say that the function (resp. set-valued function) f is closable in X × Y if there exists a subset D ⊆ X containing S and a function (resp. set-valued function) F : D → Y whose graph is equal to the closure of the set Gr f in X × Y. Such an F is called a closure of f in X × Y, is denoted by f, and necessarily extends f.
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.)
This also includes β = 1 and therefore all Lipschitz continuous functions on a bounded set are also C 0,α Hölder continuous. The function f(x) = x β (with β ≤ 1) defined on [0, 1] serves as a prototypical example of a function that is C 0,α Hölder continuous for 0 < α ≤ β, but not for α > β.