= Lacunary scaling-phase regularity counterexample
Let
$$
\theta(t)=\sum_{n\ge0}2^{-n}\sin(2^nt),\qquad a(t)=e^{i\theta(t)}.
$$
The <Fourier coefficients> of $\theta$ are absolutely summable, so $a$ belongs to the <Wiener algebra>, as does $a^{-1}$. The identity $\theta(2t)=\theta(t)+\theta(t+\pi)$ implies $a(2t)=a(t)a(t+\pi)$. Starting with a compactly supported <Daubechies wavelet> having $p\ge3$ <vanishing moments>, the <periodic phase change of a scaling function> gives $m(t)=a(t+\pi)m_0(t)$, while the canonical high-pass construction leaves $\widehat\psi(2t)=e^{-it}\overline{m_0(t+\pi)}\widehat\varphi_0(t)$ unchanged.
At a dyadic point $t_0=2\pi k/2^j$, the terms of the <difference quotient> with $n\ge j$ and $2^n|h|\le1$ each contribute $1+O((2^nh)^2)$. Their number tends to infinity, their total error is bounded, and the remaining tail contributes a bounded amount. Hence $(\theta(t_0+h)-\theta(t_0))/h=\log_2(1/|h|)+O(1)$, so neither $\theta$ nor $a$ has a finite <derivative> there. These points are <dense>. Away from the isolated zero of $m_0$ near $\pi$, multiplication by its nonzero <smooth> value cannot remove this nondifferentiability. Thus the new <MRA low-pass filter> is not <differentiable> throughout any neighborhood of $\pi$, despite integrability of the <scaling function> and unchanged <vanishing moments>. Ordinary higher <derivatives> at $\pi$ cannot be inferred, although the corresponding <Peano zero> survives.
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