I am trying to find a function continuous on $[0, 1]$ which is nowhere Hölder continuous for $0 \leq \alpha < 1$. From Wikipedia we have the example:
\begin{equation} f = \left\{ \begin{array}{rcl} 0 & \mbox{for}& x = 0 \\ \frac{1}{\ln(x)} & \mbox{for} & x > 0\end{array}\right. \end{equation}
According to Wikipedia, this function is not Hölder continuous for any $\alpha$, on the interval $[0, \frac{1}{2}]$, so my assumption was to simply use the same function, but with the non-zero map given by $f(x) = \frac{1}{\ln(\frac{x}{2})}$. However, on closer inspection the original function does not agree with the condition I am using to check if a function is not Hölder continuous, so I am wondering what part of my reasoning is incorrect.
To check if a function is not Hölder continuous, I am attempting to check if the following limit holds as $x \rightarrow 0$.
\begin{equation} \frac{|f(t+x) - f(t)|}{|x|^\alpha} \rightarrow \infty \end{equation}
With $t \in [0, 1]$, but my attempts to evaluate the limit at various points have given me a limit of $0$. Under this condition, is the function in the Wikipedia article still not Hölder continuous, and is my attempt to fit the function to the interval $[0,1]$ valid? If anyone can clarify where I'm getting lost I would appreciate it