$ \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 Characterization of Continuity

Let $A\subseteq\mathbb{R}^n$ and let $f:A\to\mathbb{R}^m$. For any point $\vec{a}\in A\cap A^a$, the function $f$ is continuous at $\vec{a}$ if and only if $\lim_{\vec{x}\to\vec{a}}f(\vec{x})=f(\vec{a})$.

Concepts

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If

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Only If

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Proof

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Alternate Characterization of Continuity

Let $A\subseteq\mathbb{R}^n$ and let $f:A\to\mathbb{R}^m$. For any point $\vec{a}\in A\cap A^a$, the function $f$ is continuous at $\vec{a}$ if and only if $\lim_{\vec{x}\to\vec{a}}f(\vec{x})=f(\vec{a})$.

Concepts

Coming soon

If

Coming soon

Only If

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Alternate Characterization of Continuity

Let $A\subseteq\mathbb{R}^n$ and let $f:A\to\mathbb{R}^m$. For any point $\vec{a}\in A\cap A^a$, the function $f$ is continuous at $\vec{a}$ if and only if $\lim_{\vec{x}\to\vec{a}}f(\vec{x})=f(\vec{a})$.

Concepts

Coming soon

If

Coming soon

Only If

Coming soon

Proof

Coming soon

Alternate Characterization of Continuity

Let $A\subseteq\mathbb{R}^n$ and let $f:A\to\mathbb{R}^m$. For any point $\vec{a}\in A\cap A^a$, the function $f$ is continuous at $\vec{a}$ if and only if $\lim_{\vec{x}\to\vec{a}}f(\vec{x})=f(\vec{a})$.

Concepts

Coming soon

If

Coming soon

Only If

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof