We know the definition of a $\alpha$-Holder continous function $\;f:I\to\Bbb R$ (where $0<\alpha\le1$ and $I\subseteq\Bbb R$ is an interval): it is a measurable function such that $$ \sup_{0\le s<t\le1}\frac{|f(t)-f(s)|}{|t-s|^{\alpha}}<+\infty. $$
But, let us consider $\alpha=1/2-\beta\;$ for some $\beta>0\;,\;$$I=[0,1]\;,\;$ $f(x)=x^{\alpha}$ and $g(t)$ be a trajectory of the standard Brownian motion.
It's clear that $f$ is exactly $\alpha$-Holder, in the sense that the above condition is no longer true if we exceed only a few on $\alpha$: for every $\epsilon>0$ $$ \sup_{0\le s<t\le1}\frac{|f(t)-f(s)|}{|t-s|^{\alpha+\epsilon}}=+\infty. $$
BUT it's important to notice that $x=0$ is the unique patological point: if we cut out the zero by considering $f$ on $[\delta,1]$, we get a $\mathcal C^{\infty}$ function, so now we get that $$ \sup_{\delta\le s<t\le1}\frac{|f(t)-f(s)|}{|t-s|^{\alpha+\epsilon}}<+\infty. $$
At the same time, things are different for $g$: every point of the trajectories of the BM is patological, in sense that, for every $J\subseteq I$ such that $\operatorname{Acc}_I(J)\neq\emptyset$ (i.e. $J$ has accumulation points in $I$) we have that, taking $\epsilon=\beta$, $$ \sup_{s<t\\s,t\in J}\frac{|g(t)-g(s)|}{|t-s|^{\alpha+\beta}}=+\infty. $$
How can we distinguish these different behaviors of functions which are, equally $\alpha-$Holder continuous? Does exists any suitable definition which highlights the subtle differences of the above functions?