Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E$ be a real inner product space, with its inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ and zero vector $0_{E}$ as fixed there, let $\mathrm{Sym}(E)$ be the set of bounded symmetric bilinear forms on $E$, and let $d_{\mathbb{R}}$ be the metric on $\mathbb{R}$ fixed there. Let $U\subseteq E$ be open in $(E,d)$ and let $u:U\to\mathbb{R}$. That $u$ is differentiable at a point of $U$ or differentiable on $U$, and its gradient $Du(x)$, are as defined there; that a form is a second derivative of $u$ at a point is as defined there; and the classes $C^{1}(U)$ and $C^{2}(U)$ are as defined there. For a subset $U'\subseteq U$, $u|_{U'}$ denotes the restriction of $u$ to $U'$. Local maxima and local minima of $u$ relative to $U$, and continuity of $u$ on $U$ as a map into $(\mathbb{R},d_{\mathbb{R}})$, are always understood with the ambient metric space $(E,d)$.

Then the following hold.

1. (Local increment bound)¶ Suppose $u$ is differentiable at $x\in U$ and let $\varepsilon\in\mathbb{R}$ be positive. Then there is a positive $\delta\in\mathbb{R}$ such that every $z\in E$ with $|z|<\delta$ satisfies $x+z\in U$ and

2. (Continuity)¶ If $u$ is differentiable at $x\in U$, then $u$ is continuous at $x$ relative to $U$. Consequently, if $u$ is differentiable on $U$ then $u$ is continuous on $U$; in particular every member of $C^{1}(U)$ and every member of $C^{2}(U)$ is continuous on $U$.

3. (First-order condition at a local extremum)¶ Suppose $u$ is differentiable at $x\in U$ and has either a local maximum or a local minimum at $x$ relative to $U$. Then $Du(x)=0_{E}$.

4. (Restriction to a smaller open set)¶ Let $U'\subseteq U$ be open in $(E,d)$ and let $x\in U'$. If $u$ is differentiable at $x$ with gradient $p$, then $u|_{U'}$ is differentiable at $x$ with gradient $p$. If $u$ is differentiable on $U$, then $u|_{U'}$ is differentiable on $U'$ with $D(u|_{U'})(y)=Du(y)$ for every $y\in U'$. If $b\in\mathrm{Sym}(E)$ is a second derivative of $u$ at $x$, then $b$ is a second derivative of $u|_{U'}$ at $x$. If $u\in C^{1}(U)$ then $u|_{U'}\in C^{1}(U')$, and if $u\in C^{2}(U)$ then $u|_{U'}\in C^{2}(U')$ with $D^{2}(u|_{U'})(y)=D^{2}u(y)$ for every $y\in U'$.