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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 graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c). [2] In the context of a polynomial in one variable x, the constant function is called non-zero constant function because it is a polynomial of degree 0, and its general form is f(x) = c, where c is nonzero.
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.
A function on an interval satisfying the condition with α > 1 is constant (see proof below). If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.
Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the ...
By the extreme value theorem, a continuous function on a closed and bounded set obtains its extreme values, implying that / | | for some constant and ¯ (,). Thus, the function q ( z ) {\displaystyle q(z)} is bounded in C {\displaystyle \mathbb {C} } , and by Liouville's theorem, is constant , which contradicts our assumption that p ...
This theorem is also valid for Hölder continuous functions, that is, if : is Hölder continuous function with constant less than or equal to , then can be extended to a Hölder continuous function : with the same constant. [4]
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1). A contraction mapping has at most one fixed point.