Is every bounded continuous function uniformly continuous?
Examples and counterexamples Every Lipschitz continuous map between two metric spaces is uniformly continuous. In particular, every function which is differentiable and has bounded derivative is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous.
Is every bounded function is continuous?
By the boundedness theorem, every continuous function on a closed interval, such as f : [0, 1] → R, is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
Does uniformly continuous imply bounded?
Each uniformly-continuous function f : (a, b) → R, mapping a bounded open interval to R, is bounded. Indeed, given such an f, choose δ > 0 with the property that the modulus of continuity ωf (δ) < 1, i.e., |x − y| < δ =⇒ |f(x) − f(y)| < 1.
Which function is bounded but not continuous?
The function f: [0, 1)→ R defined by f (x) = x is continuous and bounded but does not attain its least upper bound of 1. The function f: [0, ∞)→ R defined by f (x) = x is continuous but not bounded.
Which one is not uniformly continuous?
If f is not uniformly continuous, then there exists ϵ0 > 0 such that for every δ > 0 there are points x, y ∈ A with |x − y| < δ and |f(x) − f(y)| ≥ ϵ0. Choosing xn,yn ∈ A to be any such points for δ = 1/n, we get the required sequences.
Which function is not continuous everywhere?
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.
When is a continuous function on a compact set uniformly continuous?
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.
Why does uniform continuity of a function imply that?
If A is a bounded subset of R and f: A → R is uniformly continuous on A, then f must be bounded on A. Since it is uniformly continuous, the function is a Lipschitz function. | f ( x) − f ( y) | ≤ L | x − y |. Since A is bounded, | x − y | does not get arbitrarily big and that too is bounded by a constant. Let | x − y | ≤ M.
What is the image of a totally bounded subset under a uniformly continuous function?
The image of a totally bounded subset under a uniformly continuous function is totally bounded.
When is a function called uniformly continuous in a metric space?
Definition for functions on metric spaces. Given metric spaces (X, d 1) and (Y, d 2), a function f : X → Y is called uniformly continuous if for every real number ε > 0 there exists real δ > 0 such that for every x, y ∈ X with d 1(x, y) < δ, we have that d 2(f(x), f(y)) < ε.
