Let $f=\sqrt{|x|} \in \text{Lip}_\alpha(T)$, where $\text{Lip}_\alpha(T)$ is the set of Lipschitz function with Lipschitz constant $\alpha=1/2$ on the unit circle $T$. What is $ \|f\|=\sup_{t\in T,h \ne 0} \frac{|f(t+h)-f(t)|}{|h|^\alpha} \text{ ?} $
I need to evaluate this supremum to show that translation is not continuous in $\text{Lip}_\alpha(T)$, i.e., $ \lim_{c \to 0} \|f(\cdot)-f(\cdot+c)\| \ne 0. $