Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $V\subseteq\mathbb{R}^{n}$ be open. Subsets of $\mathbb{R}^{n}$ are regarded as subsets of the metric space $(\mathbb{R}^{n},d_{E})$, the real line is regarded as the metric space $(\mathbb{R},d_{\mathbb{R}})$ with the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line, and $\mathcal{S}(n)$ carries the metric $d_{\mathcal{S}(n)}$ of clause Second-Order Equations on Euclidean Open Sets §matrices; continuity relative to $V$ is understood with respect to these metrics. For a function $\psi:V\to\mathbb{R}$ all of whose partial derivatives exist at every point of $V$, $D\psi(y)$ denotes the gradient of $\psi$ at $y$, whose $j$th coordinate is $\partial_{j}\psi(y)$.

Then the following hold.

1. (The value map)¶ Let $k$ be a natural number with $1\le k$ and let $\psi:V\to\mathbb{R}$ be of class $C^{k}$ on $V$. Then $\psi$ is continuous at every point of $V$ relative to $V$, as a map into $(\mathbb{R},d_{\mathbb{R}})$.

2. (The gradient map of a function of class $C^{1}$)¶ Let $\psi:V\to\mathbb{R}$ be of class $C^{1}$ on $V$. Then the map $V\to\mathbb{R}^{n}$ whose value at $y$ is $D\psi(y)$ is continuous at every point of $V$ relative to $V$, as a map into $(\mathbb{R}^{n},d_{E})$.

3. (The gradient and Hessian maps of a function of class $C^{2}$)¶ Let $\varphi:V\to\mathbb{R}$ be of class $C^{2}$ on $V$. Then the map $V\to\mathbb{R}^{n}$ whose value at $y$ is $D\varphi(y)$ is continuous at every point of $V$ relative to $V$, and the map $V\to\mathcal{S}(n)$ whose value at $y$ is the Hessian matrix $D^{2}\varphi(y)$ is continuous at every point of $V$ relative to $V$, as a map into $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$.