This question arose when thinking about $f$ is a real function and it is $\alpha$-Holder continuous with $\alpha>1$. Is $f$ constant?
Example:
- The Hausdorff-dimension of the von Koch snowflake is $1/p = \frac{\log4 }{\log 3} = \log_3 4 $,
- It is the image of a $p$-Hölder-continuous path (see page 567 here).
Is the statement true or is it it missing some additional condition (which)?
