Differentiability at a Point for Maps Between Euclidean Spaces

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, and let $U$ be an open subset of Euclidean space $\mathbb{R}^n$. Let $f:U\to\mathbb{R}^m$, let $a\in U$, and let $A$ be a real matrix with $m$ rows and $n$ columns.

Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, used both on $\mathbb{R}^n$ and on $\mathbb{R}^m$; write $x+y$ for the sum of points of $\mathbb{R}^n$, $z-z'$ for the difference of points of $\mathbb{R}^m$, and $Av$ for the matrix-vector product.

We say that $f$ is differentiable at $a$ with derivative matrix $A$ if for every real $\varepsilon$ with $0<\varepsilon$ there exists a real $\delta$ with $0<\delta$ such that every $h\in\mathbb{R}^n$ with $0<\lVert h\rVert<\delta$ satisfies $a+h\in U$ and

$$\lVert f(a+h)-f(a)-A\,h\rVert\le\varepsilon\,\lVert h\rVert .$$

We say that $f$ is differentiable at $a$ if it is differentiable at $a$ with derivative matrix $A$ for some such matrix $A$.