Square-root-degenerate endpoint layer ($H(X)=1-(1+2\sqrt{X})e^{-2\sqrt{X}}$)

The singular perturbation equation $ϵ\sqrt{x}{u}^{\prime \prime }+u^{\prime }=0$ balances its derivatives on the width $x={ϵ}^{2}X$. The inner equation is $\sqrt{X}{U}^{\prime \prime }+U^{\prime }=0$, so $U^{\prime }=C{e}^{-2\sqrt{X}}$. 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.