Local quadratic coordinate at a simple stationary point

Put $h=x-x_0$, $G(t)=g(x_0+th)$, and $J(t)=G^{\mathrm{\prime }}(t)$. By [F1], [F2], [F4], and $g^{\mathrm{\prime }}(x_0)=0$, $J(0)=0$, $J^{\mathrm{\prime }}(t)=G^{\mathrm{\prime }\mathrm{\prime }}(t)=h^2{g}^{\mathrm{\prime }\mathrm{\prime }}(x_0+th)$, and $g(x)-g(x_0)=G(1)-G(0)={\int }_0^{1}J(t)dt$. Applying [F1] to $J$ and to $tJ(t)$, with $(tJ{)}^{\mathrm{\prime }}=J+t{J}^{\mathrm{\prime }}$, gives ${\int }_0^{1}J(t)dt={\int }_0^1(1-t)J^{\mathrm{\prime }}(t)dt$. Hence $g(x)-g(x_0)=(x-x_0{)}^2{\int }_0^1(1-t)g^{\mathrm{\prime }\mathrm{\prime }}(x_0+t(x-x_0))dt$. Set $q(x)=2{\int }_0^1(1-t)g^{\mathrm{\prime }\mathrm{\prime }}(x_0+t(x-x_0))dt$. For $0\le j\le 5$, repeated differentiation gives $q^{(j)}(x)=2{\int }_0^1(1-t)t^j{g}^{(j+2)}(x_0+t(x-x_0))dt$; each difference quotient converges uniformly in $t$ by [F3] and [F5], since the next derivative through $g^{(7)}$ is uniformly continuous on a compact neighborhood. Thus $q\in C^5$ and $q(x_0)=g^{\mathrm{\prime }\mathrm{\prime }}(x_0)$.

Put $\sigma =\mathrm{sgn}(g^{\mathrm{\prime }\mathrm{\prime }}(x_0))$. Shrink the neighborhood so $\sigma q(x)>0$, and define $v(x)=\sqrt{\sigma q(x)}>0$ by [F6]. For a fixed point $z$ in this neighborhood, $\mathrm{\mid }v(x)-v(z)\mathrm{\mid }=\mathrm{\mid }q(x)-q(z)\mathrm{\mid }\mathrm{/}(v(x)+v(z))\le \mathrm{\mid }q(x)-q(z)\mathrm{\mid }\mathrm{/}v(z)$, so $v$ is continuous at $z$. For $x\mathrm{\ne }z$, rationalization gives $(v(x)-v(z))\mathrm{/}(x-z)=\sigma (q(x)-q(z))\mathrm{/}((x-z)(v(x)+v(z)))\to \sigma q^{\mathrm{\prime }}(z)\mathrm{/}(2v(z))$. Thus $v^{\mathrm{\prime }}=\sigma q^{\mathrm{\prime }}\mathrm{/}(2v)$ exists and is continuous. Since $q^{\mathrm{\prime }}\in C^4$, repeated product and quotient rules [F2] applied to this identity inductively give $v\in C^5$. Define $\mathrm{\Phi }(x)=(x-x_0)v(x)$. Then $\mathrm{\Phi }\in C^5$, $g=g(x_0)+\sigma {\mathrm{\Phi }}^2\mathrm{/}2$, $\mathrm{\Phi }(x_0)=0$, and ${\mathrm{\Phi }}^{\mathrm{\prime }}(x_0)=\sqrt{\mathrm{\mid }{g}^{\mathrm{\prime }\mathrm{\prime }}(x_0)\mathrm{\mid }}>0$.

By continuity of ${\mathrm{\Phi }}^{\mathrm{\prime }}$, shrink to an open interval $J_0$ about $x_0$ where ${\mathrm{\Phi }}^{\mathrm{\prime }}\ge m>0$. For $x<y$ in $J_0$, [F3] gives $\mathrm{\Phi }(y)-\mathrm{\Phi }(x)={\mathrm{\Phi }}^{\mathrm{\prime }}(\xi )(y-x)\ge m(y-x)$, so $\mathrm{\Phi }$ is strictly increasing. Applying [F7] between points on either side of each $x\in J_0$ shows that $\mathrm{\Phi }(J_0)$ is open. Its inverse $\mathrm{\Psi }$ satisfies $\mathrm{\mid }\mathrm{\Psi }(u)-\mathrm{\Psi }(w)\mathrm{\mid }\le \mathrm{\mid }u-w\mathrm{\mid }\mathrm{/}m$, hence is continuous. At $u=\mathrm{\Phi }(x)$, the inverse difference quotient is the reciprocal of the difference quotient of $\mathrm{\Phi }$, and therefore ${\mathrm{\Psi }}^{\mathrm{\prime }}(u)=1\mathrm{/}{\mathrm{\Phi }}^{\mathrm{\prime }}(\mathrm{\Psi }(u))$. This is continuous and nonzero; applying [F2] and [F4] repeatedly to this identity gives $\mathrm{\Psi }\in C^5$, since ${\mathrm{\Phi }}^{\mathrm{\prime }}\in C^4$ and never vanishes. Thus $\mathrm{\Phi }$ is the asserted diffeomorphism.