Sums, Differences and Scalar Multiples of Functions Twice Differentiable at a Point

We work in the setting of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, the absolute value $|\cdot|$, and Euclidean space $\mathbb{R}^{n}$ with its sum and difference of points, scalar multiples, dot product, Euclidean norm $\lVert\,\cdot\,\rVert$ and notion of openness, the real matrices, their sums, differences and scalar multiples, and the matrix-vector product $Ph$, and the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, closed under sums, differences and scalar multiples, are all as fixed there. Twice differentiability at a point with a given first-order coefficient and Hessian is as defined there.

Let $U\subseteq\mathbb{R}^{n}$ be open, let $y\in U$, let $f,g:U\to\mathbb{R}$, let $p,p'\in\mathbb{R}^{n}$, let $B,B'\in\mathcal{S}(n)$ and let $\mu\in\mathbb{R}$. Purely as notation, $f+g$, $f-g$ and $\mu f$ denote the functions from $U$ to $\mathbb{R}$ whose values at $x\in U$ are $f(x)+g(x)$, $f(x)-g(x)$ and $\mu\,f(x)$ respectively.

Assume that $f$ is twice differentiable at $y$ with first-order coefficient $p$ and Hessian $B$, and that $g$ is twice differentiable at $y$ with first-order coefficient $p'$ and Hessian $B'$. The matrices $B+B'$, $B-B'$ and $\mu B$ lie in $\mathcal{S}(n)$ by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric. Then the following hold.

1. (Sum) ¶ $f+g$ is twice differentiable at $y$ with first-order coefficient $p+p'$ and Hessian $B+B'$.

2. (Scalar multiple) ¶ $\mu f$ is twice differentiable at $y$ with first-order coefficient $\mu p$ and Hessian $\mu B$.

3. (Difference) ¶ $f-g$ is twice differentiable at $y$ with first-order coefficient $p-p'$ and Hessian $B-B'$.