Elementary Properties of Local Slopes: Order, Negation, Bounds from Difference Quotients, Lipschitz Functions and Squared Distances

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 $|\nabla\psi|(x)$, $|\nabla^{+}\psi|(x)$, $|\nabla^{-}\psi|(x)$ and $|\nabla\psi|^{*}(x)$ be its local slope, super-slope, sub-slope and upper envelope of the slope, with $[a]_{+}=\max\{a,0\}$ and $[a]_{-}=\max\{-a,0\}$ as there; and let $\underline{\mathcal{C}}(\Omega)$ and $\overline{\mathcal{C}}(\Omega)$ be the sub-slope and super-slope test classes. For $\psi$ locally Lipschitz on $\Omega$ and $x\in\Omega$, a slope pair of $\psi$ at $x$ is one of $(|\nabla\psi|(x),g)$ with $g(a)=|a|$, $(|\nabla^{+}\psi|(x),g)$ with $g(a)=[a]_{+}$, and $(|\nabla^{-}\psi|(x),g)$ with $g(a)=[a]_{-}$.

1. (Order)¶ Let $\psi$ be locally Lipschitz on $\Omega$ and $x\in\Omega$. Then $|\nabla^{+}\psi|(x)\le|\nabla\psi|(x)$, $|\nabla^{-}\psi|(x)\le|\nabla\psi|(x)$ and $|\nabla\psi|(x)\le|\nabla\psi|^{*}(x)$.

2. (Negation)¶ Let $\psi$ be locally Lipschitz on $\Omega$ and $x\in\Omega$. Then $-\psi$ is locally Lipschitz on $\Omega$, and $|\nabla(-\psi)|(x)=|\nabla\psi|(x)$, $|\nabla^{+}(-\psi)|(x)=|\nabla^{-}\psi|(x)$ and $|\nabla^{-}(-\psi)|(x)=|\nabla^{+}\psi|(x)$.

3. (Upper bound from difference quotients)¶ Let $\psi,\varphi:\Omega\to\mathbb{R}$ be locally Lipschitz on $\Omega$, let $x\in\Omega$, let $(\sigma,g)$ be a slope pair of $\psi$ at $x$, and let $c$ be a nonnegative real. Suppose that for every real $\varepsilon>0$ there is a real $r>0$ such that

Then $\sigma\le c+|\nabla\varphi|(x)$.

4. (Lower bound from difference quotients)¶ Let $\psi,\varphi:\Omega\to\mathbb{R}$ be locally Lipschitz on $\Omega$, let $x\in\Omega$, let $(\sigma,g)$ be a slope pair of $\psi$ at $x$, and let $c$ be a real number. Suppose that for all reals $\varepsilon>0$ and $r>0$ there is $y\in\Omega$ with $0<d(x,y)<r$ and

Then $c-|\nabla\varphi|(x)\le\sigma$.

5. (Lipschitz functions)¶ Let $\psi:\Omega\to\mathbb{R}$ and let $L$ be a nonnegative real with $|\psi(y)-\psi(z)|\le L\,d(y,z)$ for all $y,z\in\Omega$. Then $\psi$ is locally Lipschitz on $\Omega$, and $|\nabla\psi|(x)\le L$ and $|\nabla\psi|^{*}(x)\le L$ for every $x\in\Omega$.

6. (Squared distances)¶ Suppose that $(X,d)$ has interpolation points. Let $x_{0}\in X$, let $k$ be a nonnegative real, let $C$ be a real, and let $\varphi:\Omega\to\mathbb{R}$ be given by $\varphi(x)=k\,d(x,x_{0})^{2}+C$. Then $\varphi$ is locally Lipschitz on $\Omega$,

and $\varphi\in\underline{\mathcal{C}}(\Omega)$. Moreover $-\varphi\in\overline{\mathcal{C}}(\Omega)$ and $|\nabla(-\varphi)|(x)=|\nabla^{+}(-\varphi)|(x)=2k\,d(x,x_{0})$ for every $x\in\Omega$.