Continuity of a Second-Order Equation Operator

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $U\subseteq\mathbb{R}^{n}$ be open and let $F$ be a second-order equation operator on $U$, so that $F(x,r,p,X)$ is a real number for every $x\in U$, every $r\in\mathbb{R}$, every $p\in\mathbb{R}^{n}$ and every $X\in\mathcal{S}(n)$.

1. (Continuity at a quadruple)(i) Let $x_{0}\in U$, let $r_{0}\in\mathbb{R}$, let $p_{0}\in\mathbb{R}^{n}$ and let $X_{0}\in\mathcal{S}(n)$. We say that $F$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$ if for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ such that all $y\in U$, $s\in\mathbb{R}$, $q\in\mathbb{R}^{n}$ and $Y\in\mathcal{S}(n)$ satisfying

$$d_{E}(y,x_{0})<\delta,\qquad |s-r_{0}|<\delta,\qquad \lVert q-p_{0}\rVert<\delta,\qquad d_{\mathcal{S}(n)}(Y,X_{0})<\delta$$

also satisfy

$$\bigl|F(y,s,q,Y)-F(x_{0},r_{0},p_{0},X_{0})\bigr|<\varepsilon .$$

2. (Continuity)(ii) We say that $F$ is continuous if it is continuous at $(x_{0},r_{0},p_{0},X_{0})$ for every $x_{0}\in U$, every $r_{0}\in\mathbb{R}$, every $p_{0}\in\mathbb{R}^{n}$ and every $X_{0}\in\mathcal{S}(n)$.