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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).
For instance, the function f : N → R such that f(n) := n 2 is uniformly continuous with respect to the discrete metric on N, and its minimal modulus of continuity is ω f (t) = +∞ for any t≥1, and ω f (t) = 0 otherwise. However, the situation is different for uniformly continuous functions defined on compact or convex subsets of normed ...
For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m. As a consequence, if f is continuous on [,] and () and () differ in sign, then, at some point [,], must equal zero.
Let X denote the real numbers ℝ with the usual Euclidean topology and let Y denote ℝ with the indiscrete topology (where note that Y is not Hausdorff and that every function valued in Y is continuous). Let f : X → Y be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : X → Y is continuous but its graph is not closed in X × Y. [4]
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 ...
Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.
A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over [0, π/2), x 2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can
The C 0 function f (x) = x for x ≥ 0 and 0 otherwise. The function g ( x ) = x 2 sin(1/ x ) for x > 0 . The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } with f ( x ) = x 2 sin ( 1 x ) {\displaystyle f(x)=x^{2}\sin \left({\tfrac {1}{x}}\right)} for x ≠ 0 {\displaystyle x\neq 0} and f ( 0 ) = 0 {\displaystyle f(0)=0 ...