## Are Holder continuous functions differentiable?

real analysis – Hölder continuous but not differentiable function – Mathematics Stack Exchange.

### What does it mean to have a continuous derivative?

A function is said to be continuously differentiable if the derivative exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.

**Are Holder continuous functions continuous?**

There are examples of uniformly continuous functions that are not α–Hölder continuous for any α. For instance, the function defined on [0, 1/2] by f(0) = 0 and by f(x) = 1/log(x) otherwise is continuous, and therefore uniformly continuous by the Heine-Cantor theorem.

**Can a derivative be continuous?**

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.

## Is the Cantor function continuous?

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure.

### Can a derivative be discontinuous?

The basic example of a differentiable function with discontinuous derivative is f(x)={x2sin(1/x)if x≠00if x=0. The differentiation rules show that this function is differentiable away from the origin and the difference quotient can be used to show that it is differentiable at the origin with value f′(0)=0.

**How do you show Holder continuity?**

1 Answer

- let p=1/α
- observe that |f′|p is an integrable function on (0,1)
- estimate |f(x)−f(y)| by the integral of |f′| over the interval between x and y.
- apply Hölder’s inequality to |f′|⋅1, raising |f′| to power p. This will yield a factor of |x−y|1/p as required.

**What is Equicontinuous family function?**

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

## What does continuous everywhere mean?

Every rational function is continuous everywhere it is defined, i.e., at every point in its domain. Its only discontinuities occur at the zeros of its denominator. Corollary: If p is a polynomial and a is any number, then lim p(x) = p(a).

### What is a C0 Hölder continuous function?

Hölder continuous. This also includes β = 1 and therefore all Lipschitz continuous functions on a bounded set are also C0,α Hölder continuous. The function f ( x) = xβ (with β ≤ 1) defined on [0, 1] serves as a prototypical example of a function that is C0,α Hölder continuous for 0 < α ≤ β, but not for α > β.

**What is an α-Hölder continuous function?**

For α > 1, any α–Hölder continuous function on [0, 1] (or any interval) is a constant. There are examples of uniformly continuous functions that are not α–Hölder continuous for any α.

**What is a Hölder continuous vector space?**

This is a locally convex topological vector space. If the Hölder coefficient is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm.

## Is the Hölder space locally convex?

The Hölder space Ck,α (Ω), where Ω is an open subset of some Euclidean space and k ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up through order k and such that the k th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space.