A function $f$ is Log-Lipschitz if there exists a constant $C >0$ such that
\begin{equation}
|f(x) - f(y)| \le C|x-y| |\log|x-y||
\end{equation}


I am trying to construct two functions with the following properties.


First function is $f\in \mathcal{C}^1((0,a])$ ,$a>0$ and log-Lipschitz continuous on $[0,a]$ such that 
\begin{equation}
\limsup\_{x \to0^+} x^q|f'(x)|=+\infty, \forall q \geq 1
\end{equation}
Second function is $g\in \mathcal{C}^1((0,a])$ continuous on $[0,a]$ but Holder continuous on $[0,a]$ for no $\alpha<1$ such that
\begin{equation}
\limsup\_{x \to0^+} x|g'(x)|<+\infty.
\end{equation}
I tried constructing but not getting through much(I came across these functions in context of log-lipschitz regularity of certain hyperbolic pdes).
Thanks in advance.



Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions
More generally: if $\omega$ is a modulus of continuity with $\omega'(0)=\infty$ there is an $\omega$-continuous, smooth function $f$ on $\mathbb{R}\_+$, with prescribed derivative $p\_k\in \mathbb{R}$ at points of a prescribed discrete subset $(x\_k)\_{k\ge1}$ of $\mathbb{R}\_+$: $f'(x\_k)=p\_k$ for all $k\ge1$. 


*Construction.* We may assume w.l.o.g. $\omega$ is a [concave modulus of continuity](https://en.wikipedia.org/wiki/Modulus_of_continuity#Special_moduli_of_continuity), so that $\omega(t)/t$ is decreasing with $\sup\_{t>0}\omega(t)/t=+\infty$ . 
 Fix a smooth function $\phi$ with support in $[-1/2,1/2]$ and with $\phi'(0)=1= \|\phi'\|\_\infty$. Define positive numbers $\delta\_k$ such that $2^k|p\_k|\le \omega(\delta\_k)/\delta\_k,$ and such that the intervals $I\_k$ of length $\delta\_k$ centered at $x\_k$ are pairwise disjoint.


For any $k\ge1$ consider $\phi\_k(x):=\delta\_k p\_k \phi\big({x-x\_k\over\delta\_k}\big)$, a smooth function supported in $I\_k$ such that $\phi\_k'(x\_k)=p\_k$ and $\|\phi\_k'\|\_\infty=|p\_k|$. It satisfies $|\phi\_k(x)-\phi\_k(y)|\le2^{-k}\omega(|x-y|)$ for all $x$ and $y\in\mathbb{R}$: indeed, to check the latter, it is sufficient to look at points $x$ and $y$ both in $I\_k$, and for these points $|x-y|\le\delta\_k$, so
$$|\phi\_k(x)-\phi\_k(y)|\le |p\_k||x-y|\le 2^{-k} {\omega(\delta\_k)\over\delta\_k}|x-y|\le 2^{-k} \omega( |x-y|)$$


Consider $\sum\_{k=1}^\infty \phi\_k$, a locally finite sum in $\mathbb{R}\_+$, thus defining an $\omega$-continuous function $f\in C^\infty(\mathbb{R\_+})$, with $f'(x\_k)=p\_k$. 
$$\*$$
The other question: the function $g(x):=\exp(-|\log x|^{1/2})$ is smooth on $(0,1)$ and continuous on $[0,1]$ with $g(0)=0$. It is not Hölder at $0$ because for all $\alpha>0$, $x^\alpha=o(g(x))$ as $x\to0$. But, $2xg'(x)={g(x)|\log x|^{-1/2} }=o(1)$ as $x\to0$.


