Approximability by Test Data Under Negation

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation is in force in a dimension $n$, a natural number with $1\le n$. That a quadruple is approximable by test data from above or from below for a function is as defined there.

Let $U\subseteq\mathbb{R}^{n}$ be open, let $u:U\to\mathbb{R}$, and let $-u:U\to\mathbb{R}$ be the function whose value at $z\in U$ is $-u(z)$. Let $x_{0}\in U$, let $p\in\mathbb{R}^{n}$ and let $X\in\mathcal{S}(n)$, and write $-p$ for the scalar multiple of $p$ by $-1$ and $-X$ for the scalar multiple of $X$ by $-1$, which lies in $\mathcal{S}(n)$ by Second-Order Equations on Euclidean Open Sets §matrices. Then the following hold.

1. (From above to below)¶ The quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$ if and only if the quadruple $\bigl(x_{0},(-u)(x_{0}),-p,-X\bigr)$ is approximable by test data from below for $-u$.

2. (From below to above)¶ The quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from below for $u$ if and only if the quadruple $\bigl(x_{0},(-u)(x_{0}),-p,-X\bigr)$ is approximable by test data from above for $-u$.