Test Classes for Slope-Based Viscosity Solutions on a Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space and let $\Omega\subseteq X$ be open in $(X,d)$. For a function $\psi:\Omega\to\mathbb{R}$ locally Lipschitz on $\Omega$ and $x\in\Omega$, let $|\nabla\psi|(x)$, $|\nabla^{+}\psi|(x)$ and $|\nabla^{-}\psi|(x)$ be its local slope, super-slope and sub-slope at $x$. The restriction of $d$ to $\Omega\times\Omega$ is a metric on $\Omega$, the conditions of Metric Space being statements about points of $X$; continuity of a map $\Omega\to\mathbb{R}$ is understood from $(\Omega,d)$ to $(\mathbb{R},d_{\mathbb{R}})$.

1. (Sub-slope class)¶ $\underline{\mathcal{C}}(\Omega)$ is the set of functions $\psi:\Omega\to\mathbb{R}$ that are locally Lipschitz on $\Omega$, satisfy $|\nabla^{-}\psi|(x)=|\nabla\psi|(x)$ for every $x\in\Omega$, and for which the map $\Omega\to\mathbb{R}$, $x\mapsto|\nabla\psi|(x)$, is continuous.

2. (Super-slope class)¶ $\overline{\mathcal{C}}(\Omega)$ is the set of functions $\psi:\Omega\to\mathbb{R}$ that are locally Lipschitz on $\Omega$, satisfy $|\nabla^{+}\psi|(x)=|\nabla\psi|(x)$ for every $x\in\Omega$, and for which the map $\Omega\to\mathbb{R}$, $x\mapsto|\nabla\psi|(x)$, is continuous.