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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. $

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    @did Can you help me with the translation of continuity? I try to work $\frac{|f(t+h)-f(t+c+h)-f(t)+f(t+c)|}{|h|^\alpha}$, but I get nowhere.2012-09-24

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Notice that any translated copy of square root is differentiable at $0$. Then prove the following more general statement: if $g$ is a function differentiable at $0$, then $\lim_{h\to0 } \frac{(\sqrt{|h|}+g(h)) -g(0) }{\sqrt{|h|} }=1$. Deduce that the $Lip_{1/2}$ norm of $\sqrt{|\cdot |}+g$ is at least $1$.