= Square-root-degenerate endpoint layer
{title2=$H(X)=1-(1+2\sqrt X)e^{-2\sqrt X}$}
The <singular perturbation> equation $\epsilon\sqrt x\,u''+u'=0$ balances its derivatives on the width $x=\epsilon^2X$. The inner equation is $\sqrt X\,U''+U'=0$, so $U'=Ce^{-2\sqrt X}$. Its solution with $U(0)=0$ and $U(\infty)=1$ is the displayed $H$, since $\int_0^\infty e^{-2\sqrt q}dq=1/2$. Near zero $H=2X-8X^{3/2}/3+O(X^2)$, so the first derivative is finite while the second is singular. Treat the differential equation on the open interval and impose the endpoint value by continuous extension.
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