Prove function is continuous, multivariable

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SUMMARY

The function f defined by f(x) = c.x, where c is a vector and x is a vector in ℝn, is continuous on the n-dimensional vector space Vn. To prove continuity, one must refer to the definitions provided in the relevant textbook, particularly the definition of continuity in the context of vector spaces. The continuity of linear functions, such as the one presented, follows from the properties of vector spaces and norms.

PREREQUISITES
  • Understanding of vector spaces, specifically n-dimensional vector spaces (Vn).
  • Familiarity with the concept of continuity in mathematical analysis.
  • Knowledge of linear functions and their properties.
  • Ability to interpret mathematical definitions from textbooks.
NEXT STEPS
  • Review the definitions of continuity in vector spaces from your textbook.
  • Study the properties of linear transformations and their continuity.
  • Explore norm functions and their role in defining continuity in vector spaces.
  • Practice proving continuity for various linear functions in ℝn.
USEFUL FOR

Students of mathematics, particularly those studying multivariable calculus or linear algebra, as well as educators seeking to clarify concepts of continuity in vector spaces.

ohlala191785
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Problem: If c is in Vn, show that the function f given by f(x) = c.x (c dot x, where both c and x are vectors) is continuous on ℝn.

How do I go about proving this? I'm not sure if c is supposed to be a constant or a constant vector, but since it is bolded in the book I am assuming it is a vector. Also, I'm not sure what Vn is. Finally, how do I show it is continuous?

As you can probably see, I am very lost. Any help would be appreciated.
 
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"I'm not sure what Vn is."
--
Nor am I.
However, your book has most probably defined it, say as an n-dimensional vector space with a norm function associated with it.

So, your first exercise is to scrutinize your book for the DEFINITIONS of the terms used in the actual exercise.
 

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