Weierstrass function ($g_{a,b}(x)={\sum }_{k\ge 0}{a}^{-k}\cos (b^{k}x)$)

A standard Weierstrass function is a uniformly convergent lacunary cosine series, here parameterized by $a>1$ and an integer frequency multiplier $b>1$. Its continuity follows from the Weierstrass M-test. In the range $1<a<b$, its modulus of continuity has the fractional-power scale ${\delta }^{\log a/\log b}$ for the positive odd-frequency model; the critical case $a=b$ can have the larger scale $\delta \log (1/\delta )$. These examples distinguish small approximation error from ordinary Lipschitz continuity.

For $g(x)={\sum }_{k\ge 0}{5}^{-k}\cos (5^{k}x)$, all frequencies satisfy $5^k\equiv 1(\mathrm{mod}4)$. At $x=\pi /2$, the increment is $-{\sum }_{k\ge 0}{5}^{-k}\sin (5^{k}h)$. If $5^m\le 1/h<5^{m+1}$, the first $m+1$ terms have $0<5^{k}h\le 1$ and contribute at least $(m+1)h\sin 1$, while the remaining tail has magnitude at most $5^{-m}/4\le 5h/4$. This proves a lower modulus of continuity bound of order $h\log (1/h)$ for small $h$. The inverse theorem for trigonometric approximation supplies the matching upper bound because the positive lacunary trigonometric series has $E_n(g)\asymp n^{-1}$.