Is there a continuous function $f:\left [0,\dfrac{1}{2}\right ]\to \mathbb{R}$ for which there is no constant$c>0$ such that:
$$|f(x)-f(y)|\leq\dfrac{c}{-\ln(|x-y|)},\ \forall\ x,y\in\left [0,\dfrac{1}{2}\right ] $$
As a remark, $f$ is uniformly continuous (being continuous on a bounded and closed interval), but it can't be Lipschitz (so it can't be of class $C^1$) because the above inequality is true for some constant for any Lipschitz function (easy to prove).
So we can hope to obtain at most a function $f$ that is differentiable on $\left (0,\dfrac{1}{2}\right )$. Is it possible to find one? I tried, but I didn't succeed...