Geodesic
Functions of bounded $p$-variation with given order of modulus of $p$-continuity
Matematičeskie zametki, Tome 12 (1972) no. 5, pp. 523-530.

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We construct continuous functions for which the modulus of $p$-continuity tends to zero with given order in Wiener's sense $$ V^p(\delta;f)=\sup\sum_i|f(x_i)-f(x_{i-1})|^p\qquad(p>1) $$ (the upper bound is taken over partitions satisfying the condition $x_i-x_{i-1}\leqslant\delta$). 
@article{MZM_1972_12_5_a3,
     author = {A. P. Terekhin},
     title = {Functions of bounded $p$-variation with given order of modulus of $p$-continuity},
     journal = {Matemati\v{c}eskie zametki},
     pages = {523--530},
     publisher = {mathdoc},
     volume = {12},
     number = {5},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_5_a3/}
}
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A. P. Terekhin. Functions of bounded $p$-variation with given order of modulus of $p$-continuity. Matematičeskie zametki, Tome 12 (1972) no. 5, pp. 523-530. http://geodesic.mathdoc.fr/item/MZM_1972_12_5_a3/