Show that the map is continuous

In summary, the conversation discusses using the definition of continuity to show that the map F: R^3 → R^2 given by F(x,y,z)= ( 0.5⋅(e^(x)+x) , 0.5⋅(e^(x)-x) ) is continuous. The discussion involves using the concept of preimages and open balls to show that for a given ε, there exists a δ such that the image of the ball radius δ lies inside the ball radius ε. It is suggested to start with a point within δ of (x,y,z) and see what bounds can be put on where it maps to, using the equivalence of metrics ##l_1## and ##l_
  • #1
Interior
2
0

Homework Statement



Consider the map F: R^3 →R^2 given by F(x,y,z)= ( 0.5⋅(e^(x)+x) , 0.5⋅(e^(x)-x) ) is continuous.

Homework Equations

The Attempt at a Solution


[/B]
I want to use the definition of continuity which involves the preimage:

""A function f defined on a metric space A and with values in a metric space B is continuous if and only if f^(-1)(O) is an open subset of A for any open subset O of B."

I think that we can somehow use the concept of a ball around a given point in the image and preimage.
In our case our goal is to show that F^(-1) is open, i.e. we want to show that for some radius δ>0 we can make a an open ball around any given point x∈F^(-1). If this can be done for ∀x∈F^(-1) then F^(-1) is open.

a) A ball around a point P(x,y,z)=(a,b,c) in R^3 is given by (x-a)^2 + (y-b)^2 + (z-b)^2 < δ^2 .
b) This ball is sent to R^2 by the map F creating the ball (circle) with center in F(a,b,c) and radius ε>0.

I need help connecting the information in a) with the information in b).

Thanks.

 
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  • #2
Interior said:
I need help connecting the information in a) with the information in b).
You need to show that for a given ε you can find a δ such that the image of the ball radius δ lies inside the ball radius ε.
 
  • #3
Interior said:

Homework Statement



Consider the map F: R^3 →R^2 given by F(x,y,z)= ( 0.5⋅(e^(x)+x) , 0.5⋅(e^(x)-x) ) is continuous.

Homework Equations

The Attempt at a Solution


[/B]
I want to use the definition of continuity which involves the preimage:

""A function f defined on a metric space A and with values in a metric space B is continuous if and only if f^(-1)(O) is an open subset of A for any open subset O of B."

I think that we can somehow use the concept of a ball around a given point in the image and preimage.
In our case our goal is to show that F^(-1) is open, i.e. we want to show that for some radius δ>0 we can make a an open ball around any given point x∈F^(-1). If this can be done for ∀x∈F^(-1) then F^(-1) is open.

a) A ball around a point P(x,y,z)=(a,b,c) in R^3 is given by (x-a)^2 + (y-b)^2 + (z-b)^2 < δ^2 .
b) This ball is sent to R^2 by the map F creating the ball (circle) with center in F(a,b,c) and radius ε>0.

I need help connecting the information in a) with the information in b).

Thanks.
In ##R^n,## the metrics ##l_1## and ##l_2## are "equivalent" in the sense that there exist constants ##r, s## such that ##||x||_1 \leq r ||x||_2## and ##||x||_2 \leq s ||x||_1##. Thus, characterizing continuity using open balls or open cubes can be done interchangeably.
 
Last edited:
  • #4
Haruspex, yes I know that, but I simply don't know how to do that.
Can you give some clues?
 
  • #5
Interior said:
Haruspex, yes I know that, but I simply don't know how to do that.
Can you give some clues?
Start with a point within δ of (x,y,z) and see what bounds you can put on where it maps to.
If (x-a)^2 + (y-b)^2 + (z-b)^2 < δ^2, what can you say about |x-a| etc. individually?
 
  • #6
Interior said:
Haruspex, yes I know that, but I simply don't know how to do that.
Can you give some clues?
Look at post #3.
 

Related to Show that the map is continuous

1. What does it mean for a map to be continuous?

The continuity of a map refers to the property that small changes in the input result in small changes in the output. In other words, if we have a map that assigns each point in a domain to a point in a range, then continuity means that if we make a small change to the input point, the output point will also change by a small amount.

2. How is continuity of a map mathematically defined?

Mathematically, a map is continuous at a point if the limit of the output as the input approaches that point is equal to the output at that point. This is written as:
limx→a f(x) = f(a)

3. What are the necessary conditions for a map to be continuous?

There are two necessary conditions for a map to be continuous:
1. The map must be defined at the point in consideration.
2. The limit of the map at that point must exist and be equal to the value of the map at that point.

4. How can we show that a map is continuous?

To show that a map is continuous, we need to prove that it satisfies the necessary conditions for continuity. This can be done by using the mathematical definition and showing that the limit of the map at a given point exists and is equal to the value of the map at that point. This can be achieved by using various mathematical techniques such as epsilon-delta proofs or using the properties of continuity.

5. What are the practical applications of proving that a map is continuous?

Proving that a map is continuous is important in many fields of science and engineering, including physics, chemistry, and computer science. It allows us to make accurate predictions and models based on the behavior of the map. For example, in physics, the continuity of a function can help us determine the trajectory of a moving object, while in computer science, it is essential for creating efficient algorithms and programs. Additionally, proving continuity is also an important step in proving more complex mathematical theorems and properties.

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