1
$\begingroup$

I've seen the following concept appear quite often in mathematics:

A function $f:I\subset \mathbb{R}\to\mathbb{R}$ is said to be Hölder continuous if there are constants $\alpha$ and $M$ such that $$|f(x)-f(y)|\leq M|x-y|^\alpha$$ for all $x,y\in I$.

What are some examples of Hölder continuous functions?

$\endgroup$
2
  • $\begingroup$ Any realization of Brownian motion is Hölder continuous. $\endgroup$ Nov 17, 2016 at 1:58
  • $\begingroup$ Any continuously differentiable function or Lipschitz function is Hölder continuous -- not sure what you mean by "non-trivial". $\endgroup$
    – Batman
    Nov 17, 2016 at 2:28

3 Answers 3

4
$\begingroup$

A nice example is the Cantor function. It is Hölder continuous, with exponent $\displaystyle\alpha=\frac{\ln 2}{\ln 3}$.

$\endgroup$
4
$\begingroup$

$x^{\alpha}$ for $x>0$ and 0 otherwise for $0 < \alpha <1$ is Holder continuous of order $\alpha.$

$\endgroup$
1
$\begingroup$

Another example is $\arcsin x $ that is $1/2$ H"older continuous in $[-1,1]$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .