Twice Differentiability at a Point

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, and the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product and notion of openness, are as fixed there. Write $\mathcal{S}(n)$ for the set of symmetric real $n\times n$ matrices and $Bh$ for the matrix-vector product.

Let $U\subseteq\mathbb{R}^{n}$ be open, let $f:U\to\mathbb{R}$, let $y\in U$, let $p\in\mathbb{R}^{n}$ and let $B\in\mathcal{S}(n)$.

Definition. (i) The function $f$ is twice differentiable at $y$ with first-order coefficient $p$ and Hessian $B$ if for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $h\in\mathbb{R}^{n}$ with $\lVert h\rVert<\delta$ satisfies $y+h\in U$ and

$$\Bigl|f(y+h)-f(y)-p\cdot h-\tfrac{1}{2}\,h\cdot(Bh)\Bigr|\le\varepsilon\,\lVert h\rVert^{2}.$$

The function $f$ is twice differentiable at $y$ if it is twice differentiable at $y$ with first-order coefficient $p$ and Hessian $B$ for some $p\in\mathbb{R}^{n}$ and some $B\in\mathcal{S}(n)$. By A Symmetric Matrix is Determined by its Quadratic Form, and a Second-Order Expansion by its Coefficients (ii) there is then exactly one such pair, so the notation

$$D^{2}f(y)=B$$

for the Hessian of $f$ at $y$ is unambiguous. (ii)