One example of a continuous nowhere differentiable function whose modulus of continuity does not exceed a given one

There exist positive numbers $C$ and $c$ such that, for an arbitrary concave up function $\omega(t)$ of the modulus of continuity type with $\omega(t)/t\to+\infty$ as $t\to+0$, one can construct an example of a continuous nowhere differentiable Weierstrass-type function $W_\omega(x)$ satisfying the following conditions: $1^{\circ}$. The modulus of continuity of $W_\omega(x)$ does not exceed $C\omega(t)$. $2^{\circ}$. For each point $x_0$, there exists a sequence $\{x_n\}$ convergent to $x_0$, such that $|W_\omega(x_n)-W_\omega(x_0)|>c\,\omega(|x_n-x_0|)$ for each $n$. $3^{\circ}$. At each point $x_0$, the derivative numbers of $W_\omega(x)$ take all values from the interval $[-\infty;+\infty]$.