Geometric intepretation of Holder continuous functions?

Solution 1:

Hoelder continuity is about the roughness of a path. So there are some extremes. First of all if $f$ is $\alpha$ Hoelder continuous with $\alpha>1$, then $f$ is constant (very easy to prove).

A function that is Hoelder continuous with $\alpha=1$ is differentiable a.e.

So if you're Hoelder continuous with $\alpha\ge 1$ things are very nice. Less than $1$ and things are much less nice.

The lower your Hoelder exponent is, the rougher the path is. In particular $\alpha=\frac12$ is very critical. I do research in rough path theory, which handles the case when $\alpha<\frac12$ but this is much more advanced.

I think the best way to understand different Hoelder continuity is to look at some paths! I do work with what's called fractional Brownian motion. Fractional Brownian motion is a stochastic process that has a parameter $H$. $H$ ends up being the Hoelder exponent of the path a.s.

So look at some fractional Brownian motions! Here's a few pictures from Wikipedia that I think will really help clarify how "rough" a path is.

Also to answer your question, would a Hoelder continuous path with $\alpha=.3$ be nicer than $.2$, YES.