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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.
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.
A sphere is simply connected because every loop can be contracted (on the surface) to a point. The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center.
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 pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f ( x ) = lim n → ∞ cos ( π x ) 2 n {\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}} takes the value 1 {\displaystyle 1} when x {\displaystyle x} is an integer and 0 {\displaystyle ...
The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C (a, b) of continuous functions on (a, b), for it may contain unbounded functions.
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 ...
Like some other fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot. In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere.