Let $\alpha>0$. By considering the set $E_{m}$ consisting of those $f \in C([0,1])$ for which there exists an $x \in[0,1]$ with $|f(x+h)-f(x)| \leqslant m|h|^{\alpha}$ for all $x+h \in[0,1]$, or otherwise, give a Baire category proof of the existence of continuous functions $f$ on $[0,1]$ such that

$$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

at each $x \in[0,1]$.

Are the following statements true? Give reasons.

(i) There exists an $f \in C([0,1])$ such that

$$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

for each $x \in[0,1]$ and each $\alpha>0$.

(ii) There exists an $f \in C([0,1])$ such that

$$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

for each $x \in[0,1]$ and each $\alpha \geqslant 0$.