On functions of van der Waerden type

Abstract

Let $\omega(t)$ be an arbitrary modulus of continuity type function, such that $\omega(t)/t\to+\infty$, as $t\to+0$. We construct a continuous nowhere-differentiable function $V_\omega(x)$, $x\in[0;1]$, that satisfies the following conditions: 1)  its modulus of continuity satisfies the estimate $O(\omega(t))$ as $t\to+0$; 2) for some positive $c$ at each point $x_0$ holds $\limsup{|V_\omega(x){-}V_\omega(x_0)|}\big/{\omega(|x{-}x_0|)}>c$ as $x\to x_0$; 3) at each point $x_0$ holds $\liminf{|V_\omega(x){-}V_\omega(x_0)|}\big/{\omega(|x{-}x_0|)}=0$ as $x\to x_0$.

Keywords

• modulus of continuity
• nowhere-differentiable function
• van der waerden type function