= Solution
Let the perturbation <velocity> and <pressure> be $(u,w,p)(z)e^{ik(x-ct)}$, with real $k>0$. Linearizing the <Euler equations> about the <inviscid parallel shear flow> gives
$$
ik(U-c)u+U'w=-ikp/\rho_0,\qquad ik(U-c)w=-p'/\rho_0,\qquad iku+w'=0.
$$
Use $u=\phi'$, $w=-ik\phi$ to satisfy <incompressibility>. The first momentum equation gives $p/\rho_0=-(U-c)\phi'+U'\phi$; differentiate it and use the second equation. The <Rayleigh equation for inviscid shear flow> follows:
$$
\boxed{(U-c)(\phi''-k^2\phi)-U''\phi=0,\qquad \phi\to0\quad(z\to\pm\infty).}
$$
A growing temporal <normal mode> has $c_i=\operatorname{Im}c>0$ and growth rate $kc_i$.
<Rayleigh's inflection-point theorem> states that such an instability of a smooth parallel profile requires $U''$ to change sign somewhere. To prove it, divide by $U-c$, multiply by $\phi^*$, integrate over $z$ and integrate the second derivative by parts. Decay removes the endpoint terms, giving
$$
\int_{-\infty}^\infty(|\phi'|^2+k^2|\phi|^2)\,dz+\int_{-\infty}^\infty\frac{U''}{U-c}|\phi|^2\,dz=0.
$$
Taking the imaginary part yields
$$
c_i\int_{-\infty}^\infty\frac{U''|\phi|^2}{(U-c_r)^2+c_i^2}\,dz=0.
$$
The weighting is nonnegative. A one-signed $U''$ that is not identically zero cannot satisfy this identity for a nontrivial mode; otherwise the mode vanishes on an interval and uniqueness of the differential equation forces it to vanish throughout. If $U''\equiv0$, the real part of the identity is already a strictly positive integral for a nontrivial decaying mode. This proves the necessary sign-change condition.
<Fjørtoft's theorem> strengthens this: if $U_s$ is the velocity at an inflection point, instability requires $U''(U-U_s)<0$ somewhere. This is a necessary condition, not a sufficient criterion. For $U=\tanh z$, $U_s=0$ and
$$
U''(U-U_s)=-2\tanh^2z\,\operatorname{sech}^2z<0\quad(z\ne0).
$$
The possibility of instability is therefore consistent with <Fjørtoft's theorem>.
For the piecewise-linear approximation, $U''=0$ within each of the three regions. The decaying solutions outside and the two independent interior solutions can be written
$$
\phi=\begin{cases}A_-e^{k(z+1)},&z<-1,\\Ce^{kz}+Ee^{-kz},&-1<z<1,\\A_+e^{-k(z-1)},&z>1.\end{cases}
$$
At each corner, the normal <velocity> is continuous, so $\phi$ is continuous. <Pressure matching at a piecewise-linear shear interface> additionally gives $[(U-c)\phi'-U'\phi]=0$, hence $(U-c)[\phi']=[U']\phi$. At $z=1$ the slope jumps from $1$ to $0$; at $z=-1$ it jumps from $0$ to $1$. Substituting the decaying outer solutions gives
$$
\begin{pmatrix}1-2k+2kc&e^{-2k}\\e^{-2k}&1-2k-2kc\end{pmatrix}\begin{pmatrix}C\\E\end{pmatrix}=0.
$$
Thus the <inviscid instability of a piecewise-linear mixing layer> has <dispersion relation>
$$
\boxed{4k^2c^2=(2k-1)^2-e^{-4k}.}
$$
At $k=1/2$, $c=\pm i/e$, so the root with positive imaginary part is an explicitly growing mode, with \b[growth rate $1/(2e)$]. More generally the unstable interval is $0<k<k_c$, where the unique $k_c>1/2$ solves $2k_c-1=e^{-2k_c}$. This direct calculation establishes instability of the approximation; passing a necessary inflection test alone would not have established it.
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