The one-dimensional action variable of a family of two-turning-point energy levels

Let $J_0\subseteq \mathbb{R}$ be an open interval, let $U:J_0\to \mathbb{R}$ be continuous (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point), let $m>0$, let $E_0<E_1$ be reals, and let $a,b:(E_0,E_1)\to J_0$ be functions with

$a(E)<b(E),U(a(E))=U(b(E))=E,U(q)<E(a(E)<q<b(E))(E_0<E<E_1)\mathrm{.}$

The action variable of this datum is the function $J_{\mathrm{a}}:(E_0,E_1)\to \mathbb{R}$,

$J_{\mathrm{a}}(E):=\frac{1}{\pi }{\int }_{a(E)}^{b(E)}\sqrt{2m(E-U(q))}dq,$

the integral being the Riemann integral on the compact interval $[a(E),b(E)]$ (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$). It is defined because $q\mapsto \sqrt{2m(E-U(q))}$ is continuous on $[a(E),b(E)]$: the radicand is a continuous nonnegative function there (it vanishes exactly at the two endpoints). Existence and uniqueness of nonnegative roots follow from Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$. For $x,y\ge 0$, if $s=\sqrt{x}\ge t=\sqrt{y}$, then $(s-t{)}^2\le (s-t)(s+t)=x-y$; interchanging $x,y$ gives $\mathrm{\mid }\sqrt{x}-\sqrt{y}{\mathrm{\mid }}^2\le \mathrm{\mid }x-y\mathrm{\mid }$. Thus $\mathrm{\mid }x-y\mathrm{\mid }<{\epsilon }^2$ implies $\mathrm{\mid }\sqrt{x}-\sqrt{y}\mathrm{\mid }<\epsilon$, proving continuity, including at zero. The continuous integrand is Riemann integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion.

Reading of the notation. The interval $[a(E),b(E)]$ is the classically allowed segment between the two turning points of the energy $E$, and the closed curve $p=\pm \sqrt{2m(E-U(q))}$ encloses the area $A(E)=2{\int }_{a(E)}^{b(E)}\sqrt{2m(E-U(q))}dq$ (the upper graph minus the lower graph), so $J_{\mathrm{a}}(E)=A(E)\mathrm{/}(2\pi )$; no physical interpretation of $m$, $U$ or $E$ is a mathematical premise here, and no period is claimed by this definition.