Prove function is continuous, multivariable

After you have done that, you might want to tell us what you found.In summary, to prove that f(x) = c.x is continuous on ℝn, first define Vn as an n-dimensional vector space with a norm function. Then, determine whether c is a constant or a constant vector and use the definitions from the book to show its continuity.
  • #1
ohlala191785
18
0
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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  • #2
"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.
 

FAQ: Prove function is continuous, multivariable

What does it mean for a function to be continuous?

Continuous functions are those that have no sudden jumps or breaks in their graph. This means that as the input values of the function change, the output values change in a smooth and connected manner.

How can I prove that a function is continuous?

To prove that a function is continuous, you need to show that it meets the three criteria for continuity: the function is defined at the point in question, the limit of the function exists at that point, and the limit is equal to the function value at that point. You can use algebraic manipulations and theorems to show that these criteria are met.

Can a multivariable function be continuous in one variable but not the other?

Yes, it is possible for a multivariable function to be continuous in one variable but not the other. This is because continuity is dependent on the behavior of the function at a specific point, and the behavior of a multivariable function can differ along different paths or variables.

What are the consequences of a function not being continuous?

If a function is not continuous, it means that there are sudden jumps or breaks in its graph. This can lead to unpredictable and unstable behavior, making it difficult to accurately model and analyze the function.

Can a function be continuous but not differentiable?

Yes, it is possible for a function to be continuous but not differentiable at certain points. This usually occurs when there is a sharp turn or corner in the graph of the function, which makes it impossible to find a unique tangent line at that point.

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