Geodesic
One example of a continuous nowhere differentiable function whose modulus of continuity does not exceed a given one
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 4, pp. 224-233 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $[-\infty;+\infty]$.
Keywords: modulus of continuity, nowhere differentiable continuous function, derivative numbers, Weierstrass-type nowhere differentiable continuous function. 
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A. I. Rubinshtein; D. S. Telyakovskii. One example of a continuous nowhere differentiable function whose modulus of continuity does not exceed a given one. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 4, pp. 224-233. http://geodesic.mathdoc.fr/item/TIMM_2024_30_4_a17/

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