Functions of bounded $p$-variation with given order of modulus of $p$-continuity

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 $x_i-x_{i-1}\leqslant\delta$).