= Weierstrass function
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{title2=$g_{a,b}(x)=\sum_{k\ge0}a^{-k}\cos(b^kx)$}
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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>.
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