Let $f$ be a real valued function defined on the domain $[0,\infty)$, such that $f$ is strictly increasing and $f$ is continuous on $[0,\infty)$. It seems to me that, at any point $a$, in a neighborhood $U$, $f(x)-f(a)$ has to be dominated by something like $(x-a)^\alpha$, for some $\alpha>0$, at any point, i.e. I suspect that this kind of functions will be H$\ddot{\mathrm{o}}$lder continuous at any given point (maybe with different $\alpha$ s). But I cannot materialize this intuition of mine using some formal mathematical techniques.
Is it possible for $f$ not to be H$\ddot{\mathrm{o}}$lder continuous at some point? If it is not possible can anyone give me a hint to the techniques that might be helpful addressing this kind of question?
In case such an example do not exist, please do not give a full answer, but give hint to possible methods that I have to learn so that I can address this question myself. Any help is appreciated.
Edit: By the way, note that at any point where the functions is differentiable and the derivative is finite, the functions is automatically Lipscitz continuous at that point. Also, I suppose, though I am not sure, the points where the functions has one of its Dini derivatives finite, it is Lipscitz at that point. Thus the main points of interest are the points where the function is not differentiable and also the Dini derivatives blow up, which I think leave us the only possibility as the point $0$.
Answer
Hint: Try something like
$$f(x) = \sum_{n=1}^{\infty}c_n x^{1/n}$$
for suitable positive constants $c_n.$
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