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, 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 $f'(x_k)=p_k$.
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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 $x\to0$.
