There exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable

We verify that there exist no gaps between Gevrey differentiable and nowhere Gevrey differentiable in the sense that for given $s>1$, there is a nowhere Gevrey differentiable function on $\mathbb{R}$ of order $s$ that is Gevrey differentiable of order $r$ for any $r>s$, which also gives a strong example that is Gevrey differentiable but nowhere analytic.