Local Slope, Super-Slope, Sub-Slope and Upper Slope Envelope of a Locally Lipschitz Function on a Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $\Omega\subseteq X$ be open in $(X,d)$, let $B_{d}(x,r)$ be the open ball with center $x\in X$ and radius $r>0$, and let $\psi:\Omega\to\mathbb{R}$ be locally Lipschitz on $\Omega$. For $a\in\mathbb{R}$ let $[a]_{+}=\max\{a,0\}$ and $[a]_{-}=\max\{-a,0\}$ be maxima of two elements; then $0\le[a]_{+}\le|a|$ and $0\le[a]_{-}\le|a|$ by claims 1 and 3 of Elementary Properties of the Maximum of Two Elements and claim 3 of Properties of the Absolute Value in an Ordered Field. Let $G$ be the set of the three maps $g:\mathbb{R}\to\mathbb{R}$ given by $g(a)=|a|$, $g(a)=[a]_{+}$ and $g(a)=[a]_{-}$; each takes nonnegative values not exceeding $|a|$.

Fix $x\in\Omega$. By Locally Lipschitz Function on an Open Subset of a Metric Space §locally-lipschitz there are a real $r_{0}>0$ with $B_{d}(x,r_{0})\subseteq\Omega$ and a real $L\ge0$ with $|\psi(y)-\psi(z)|\le L\,d(y,z)$ for all $y,z\in B_{d}(x,r_{0})$; we call such a pair $(r_{0},L)$ a Lipschitz pair at $x$.

For $g\in G$ and real $r>0$ let

a set of nonnegative reals containing $0$, and let $\rho^{g}(x)$ be the set of reals $r>0$ for which $A^{g}_{r}(x)$ is bounded above. For a Lipschitz pair $(r_{0},L)$ at $x$, every element of $A^{g}_{r_{0}}(x)$ is at most $L$, so $r_{0}\in\rho^{g}(x)$. Hence, by The Real Numbers: Standing Notation and Background §bounds, $\sup A^{g}_{r}(x)$ exists and is nonnegative for every $r\in\rho^{g}(x)$, and the nonempty set $\{\sup A^{g}_{r}(x):r\in\rho^{g}(x)\}$ is bounded below by $0$, so that

exists and satisfies $0\le S^{g}\psi(x)$.

1. (Local slope)¶ The local slope of $\psi$ at $x$ is $|\nabla\psi|(x)=S^{g}\psi(x)$ for $g(a)=|a|$.

2. (Super-slope)¶ The super-slope of $\psi$ at $x$ is $|\nabla^{+}\psi|(x)=S^{g}\psi(x)$ for $g(a)=[a]_{+}$.

3. (Sub-slope)¶ The sub-slope of $\psi$ at $x$ is $|\nabla^{-}\psi|(x)=S^{g}\psi(x)$ for $g(a)=[a]_{-}$.

4. (Upper envelope of the slope)¶ For real $r>0$ let $E_{r}(x)=\{|\nabla\psi|(y):y\in\Omega,\ d(x,y)<r\}$, a set of nonnegative reals containing $|\nabla\psi|(x)$, and let $\rho^{*}(x)$ be the set of reals $r>0$ for which $E_{r}(x)$ is bounded above. For a Lipschitz pair $(r_{0},L)$ at $x$ and $y\in B_{d}(x,r_{0}/2)$, the triangle inequality gives $B_{d}(y,r_{0}/2)\subseteq B_{d}(x,r_{0})$, so every element of $A^{g}_{r_{0}/2}(y)$ with $g(a)=|a|$ is at most $L$ and $|\nabla\psi|(y)\le L$; thus $r_{0}/2\in\rho^{*}(x)$. The upper envelope of the slope of $\psi$ at $x$ is

which exists by The Real Numbers: Standing Notation and Background §bounds and satisfies $0\le|\nabla\psi|^{*}(x)$, every $\sup E_{r}(x)$ being at least the nonnegative number $|\nabla\psi|(x)$.