Can somebody provide an example of a function that is continuous on [0,1] but nowhere Holder continuous with degree $\alpha$? Why is the function continuous but nowhere Holder continuous?
By nowhere Holder continuous, I mean that
$\frac{|f(x + t_n) - f(x)|}{|t_n|^\alpha} \rightarrow +\infty$ where $t_n$ is a sequence which $\rightarrow 0 $ as $n\rightarrow +\infty$.
I'll denote $C([0,1])$ by $C.$ For $j,k,l \in \mathbb N,$ let $E_{jkl}$ denote the set of functions $f\in C$ for which there exists $a\in [0,1]$ such that $|f(x)-f(a)| \le j|x-a|^{1/k}$ in the neighborhood $(a-1/l,(a+1/l) \cap [0,1].$
Claim: Each $E_{jkl}$ is closed and nowhere dense in $C.$
If the claim is proved, then because $C$ is a complete metric space, Baire shows there are functions $g\in C$ that belong to no $E_{jkl}.$ For any such $g,$ the following is true: If $\alpha> 0,$ $a\in [0,1],$ $C>0,$ and $U$ is a neighborhood of $a$ relative to $[0,1],$ then
$$|g(x)- g(a)| \le C|x-a|^\alpha$$
fails to hold for some $x\in U.$ This $g$ is thus "nowhere Holder" in a very strong sense. (You never defined "nowhere Holder", btw.)
So you have the claim left to verify.