= Jump response of a localized analyzing function
{title2=$c_\psi(H(\cdot-x_0);a,b)=\sqrt a\int_{(x_0-b)/a}^{\infty}\overline{\psi(u)}\,du$}
Let $\psi$ be an <integrable function> with at least one <vanishing moment>. For a unit <jump discontinuity> $f(x)=H(x-x_0)$, where $H$ is the <Heaviside step function>, substitution into the coefficient gives
$$
c_\psi(f;a,b)=\sqrt a\int_\theta^\infty\overline{\psi(u)}\,du,\qquad\theta=\frac{x_0-b}{a}.
$$
The signed <integral> of the whole analyzing <function> is zero, but the <jump discontinuity> cuts through its <support of a function> and usually leaves an unbalanced partial <integral>. At fixed relative position $θ$ with nonzero partial <integral>, the coefficient decays as $\sqrt a$, slower than the $a^{q+1/2}$ <Taylor remainder> bound on a <smooth> region. “Edges survive” means this relative contrast: the normalized coefficients still tend to zero as $a\to0$.
For an original exact comparison, use the <smooth> <polynomial> $f(x)=x+x^2$ at $b=0$. The <Haar wavelet> $h$ and the <second-difference analyzing function> $η$ give
$$
c_h(f;a,0)=-\frac{a^{3/2}+a^{5/2}}4,\qquad c_\eta(f;a,0)=\frac2{\sqrt6}a^{5/2}.
$$
The linear term cancels for $η$; it remains for $h$. For the fixed unit <jump discontinuity> $H(x)$, choose $b=-a/4$ so that $θ=1/4$ at every scale. Then
$$
c_h(H;a,-a/4)=-\frac{\sqrt a}{4},\qquad c_\eta(H;a,-a/4)=-\frac{\sqrt a}{4\sqrt6}.
$$
\Image[/wavelet-smooth-jump-scale-coefficients.png]
{title=Exact normalized detail coefficients across scales for a smooth polynomial and a fixed unit jump}
Position matters. If an analyzing <function>'s <support of a function> lies entirely on one constant side of the <jump discontinuity>, its coefficient is zero. Even a crossing can cancel: $η$ has zero partial <integral> at $θ=3/2$. A <jump discontinuity> on a <dyadic interval> boundary can likewise escape finer <Haar wavelets> whose <supports> do not cross it. These examples describe particular <functions> and normalizations, not a universal ranking of <wavelets> for <signal processing>.
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