Difference between revisions of "Continuous Vector Functions"

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(Created page with "http://mathonline.wikidot.com/continuity-of-vector-valued-functions")
 
 
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http://mathonline.wikidot.com/continuity-of-vector-valued-functions
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Like with continuity of single variable real-valued functions, the continuity of vector-valued functions is as follows:
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: '''Definition:''' Let <math>\vec{r}(t)</math> be a vector-valued function. We say that <math>\vec{r}(t) = (x(t), y(t), z(t))</math> is '''Continuous at <math>a</math>''' if <math>\lim_{t \to a} \vec{r}(t) = \vec{r}(a)</math>, that is <math>\lim_{t \to a} x(t) = x(a)</math>, <math>\lim_{t \to a} y(t) = y(a)</math> and <math>\lim_{t \to a} z(t) = z(a)</math>. We say that <math>\vec{r}(t)</math> is '''Continuous on The Interval <math>I</math>''' if <math>\vec{r}(t)</math> is continuous for all <math>t \in I</math>.
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We should note that from the definition of <math>\vec{r}(t)</math> being continuous at <math>a</math>, we must have that the component functions <math>x(t)</math>, <math>y(t)</math>, and <math>z(t)</math> all be continuous at <math>a</math>.
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For example, suppose we wanted to show that <math>\vec{r}(t) = (t, -2t^2, \cos t)</math> is continuous at <math>\pi</math>. We notice that <math>x(t) = t</math>, <math>y(t) = -2t^2</math>, and <math>z(t) = \cos t</math> are all continuous real-valued functions, and so they're also all continuous at <math>\pi</math>. Therefore, we have that:
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<div style="text-align: center;"><math> \begin{align} \quad \quad \lim_{t \to \pi} \vec{r}(t) = \left ( \lim_{t \to \pi} t, \lim_{t \to \pi} -2t^2, \lim_{t \to \pi} \cos t \right ) = (\pi, -2\pi^2, -1) = \vec{r}(\pi) \end{align}</math></div>
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==Licensing==
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Content obtained and/or adapted from:
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* [http://mathonline.wikidot.com/continuity-of-vector-valued-functions Continuity of Vector-Valued Functions, mathonline.wikidot.com] under a CC BY-SA license

Latest revision as of 09:28, 30 October 2021

Like with continuity of single variable real-valued functions, the continuity of vector-valued functions is as follows:

Definition: Let be a vector-valued function. We say that is Continuous at if , that is , and . We say that is Continuous on The Interval if is continuous for all .

We should note that from the definition of being continuous at , we must have that the component functions , , and all be continuous at .

For example, suppose we wanted to show that is continuous at . We notice that , , and are all continuous real-valued functions, and so they're also all continuous at . Therefore, we have that:


Licensing

Content obtained and/or adapted from: