Is there a function that is Holder continuous but not Lipschitz continuous?
[Math] Holder continuous but not Lipschitz
continuitymetric-spacesreal-analysis
continuitymetric-spacesreal-analysis
Is there a function that is Holder continuous but not Lipschitz continuous?
Best Answer
From wiki,
The function $f(x) = x^\beta$ (with $\beta \le 1$) defined on $[0, 1]$ serves as a prototypical example of a function that is $C^{0,\alpha}$ Hölder continuous for $0 < \alpha \le \beta$, but not for $\alpha>\beta$. In particular, if we choose $\beta<1$, it's not Lipschitz.
Do you know how to check this statement?