Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if $$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<|y|\leq \delta}|f(x+y)-f(x)|}{\delta^{a}} = \infty, $$ giving any $\epsilon>0$, is it true that $$ \varliminf_{\delta \rightarrow 0} \frac{\sup_{0<|y|\leq \delta}|f(x+y)-f(x)|}{\delta^{a+\epsilon}} = \infty $$ ?

Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if $$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \infty, $$ then, given any $\epsilon>0$, is it true that $$ \varliminf_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a+\epsilon}} = \infty? $$