$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Alternate Definition of Differentiability

Let $A\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. The function $f$ is differentiable at $\vec{a}$ if and only if there exists a linear map $l:\mathbb{R}^n\to\mathbb{R}^m$ and a function $r:A\to\mathbb{R}^m$ that is continuous at $\vec{a}$ and satisifies $r(\vec{a})=0$ such that $$\begin{align*} f(\vec{x})=f(\vec{a})+l(\vec{x}-\vec{a})+r(\vec{x})||\vec{x}-\vec{a}|| \end{align*}$$

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Proof

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Alternate Definition of Differentiability

Let $A\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. The function $f$ is differentiable at $\vec{a}$ if and only if there exists a linear map $l:\mathbb{R}^n\to\mathbb{R}^m$ and a function $r:A\to\mathbb{R}^m$ that is continuous at $\vec{a}$ and satisifies $r(\vec{a})=0$ such that $$\begin{align*} f(\vec{x})=f(\vec{a})+l(\vec{x}-\vec{a})+r(\vec{x})||\vec{x}-\vec{a}|| \end{align*}$$

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result
Concepts
If
Only If
proof

Alternate Definition of Differentiability

Let $A\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. The function $f$ is differentiable at $\vec{a}$ if and only if there exists a linear map $l:\mathbb{R}^n\to\mathbb{R}^m$ and a function $r:A\to\mathbb{R}^m$ that is continuous at $\vec{a}$ and satisifies $r(\vec{a})=0$ such that $$\begin{align*} f(\vec{x})=f(\vec{a})+l(\vec{x}-\vec{a})+r(\vec{x})||\vec{x}-\vec{a}|| \end{align*}$$

Concepts

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If

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Only If

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Proof

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Alternate Definition of Differentiability

Let $A\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. The function $f$ is differentiable at $\vec{a}$ if and only if there exists a linear map $l:\mathbb{R}^n\to\mathbb{R}^m$ and a function $r:A\to\mathbb{R}^m$ that is continuous at $\vec{a}$ and satisifies $r(\vec{a})=0$ such that $$\begin{align*} f(\vec{x})=f(\vec{a})+l(\vec{x}-\vec{a})+r(\vec{x})||\vec{x}-\vec{a}|| \end{align*}$$

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Proof

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result
concepts
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proof