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Continuous dependence of an ODE solution on parameters here means continuity of the map $\beta \mapsto y_{\beta }$ into $C([a-h,a+h])$ with its supremum norm: for every ${\beta }_0\in I$ and $\epsilon >0$, all sufficiently close $\beta \in I$ satisfySubtract the two integral equations and use the uniform Lipschitz constant $L$ on the rectangle. With $v(x)=|y_{\beta }(x)-y_{\gamma }(x)|$ we obtainPart (i), with $c=|\beta -\gamma |$ and $A=L$, therefore proves, for $L>0$,The case $\beta =\gamma$ follows from uniqueness or the same estimate with zero forcing. If $L=0$, the direct integral estimate gives $\Vert y_{\beta }-y_{\gamma }{\Vert }_{\infty }\le h|\beta -\gamma |$. In either case the map is even Lipschitz continuous, and hence continuous, throughout $I$ in the stated supremum norm.

The useful high-density core bound from TOV compactness follows without choosing the unknown high-density equation of state. Let $a$ be the radius at which the decreasing energy density first reaches ${\rho }_0$, and put $\mu =m(a)$, $p_0=p({\rho }_0)$. The Tolman–Oppenheimer–Volkoff equation gives $p^{\prime }<0$ away from a nonempty centre; since $dp/d\rho >0$, energy density decreases outward. HenceFor the local stellar compactness, if $z\ge 0$, $1-z+\sqrt{1+z}\le 2$, since its derivative is $-1+(2\sqrt{1+z}{)}^{-1}<0$. Applying the supplied compactness inequality at the core boundary therefore gives $\mu /a<4/9$. Combining the two bounds yieldsThe pressure-dependent form further restricts the allowed core-boundary data:In particular, positivity of its right side gives $6\pi a^2{p}_0<3$, so $a^2<1/(2\pi p_0)$ when $p_0>0$. These restrictions depend only on known quantities, not on the equation of state above ${\rho }_0$.

The intended stellar-mass argument now attaches the known low-density envelope to each admissible pair $(a,\mu )$ with pressure $p_0$. Its Tolman–Oppenheimer–Volkoff equation involves only the known equation of state. For a physical low-density envelope with a uniformly bounded mass on this bounded set of core data, taking the supremum of the envelope-plus-core masses gives an upper mass limit independent of any high-density continuation. Stars with ${\rho }_c\le {\rho }_0$ are also determined entirely by the known equation of state. Unknown dense-core physics cannot evade the displayed core bounds.

A qualification is necessary for a claim about the entire star: the printed monotonicity assumptions alone do not guarantee a finite maximum total mass for an arbitrary known low-density equation of state. An explicit low-density polytropic mass divergence occurs for $p=K{\rho }^{5/4}$ with fixed $K>0$. In the low-density Newtonian limit, this is a stellar polytrope of index $n=4$, whose Lane-Emden equation has a finite first zero. With $\rho ={\rho }_c{\vartheta }^4$ and $r=\ell \xi$, hydrostatic balance fixeswhere ${\mu }_4>0$ is the dimensionless Lane-Emden surface mass constant. Thus $M\to \infty$ as ${\rho }_c\to 0$, although $M/R\propto {\rho }_c^{1/4}\to 0$. The first Lane-Emden zero is simple. The continuous dependence of an ODE solution on parameters therefore preserves this finite surface for sufficiently small relativistic parameter. The relativistic correction is controlled by $p_c/{\rho }_c=K{\rho }_c^{1/4}\to 0$, so the corresponding regular finite-radius Tolman–Oppenheimer–Volkoff equation solutions have the same limiting scalings. They never sample $\rho >{\rho }_0$, and satisfy the local positivity and monotonicity assumptions in their nonvacuum interiors. This shows exactly why a suitable bounded-envelope hypothesis, or an appropriate restriction to physically stable stellar models, must accompany the intended total-mass conclusion. The compactness inequality supplies a bound on the unknown core, not a bound on the mass of every conceivable dilute envelope.