Show that the map is continuous

[Interior]

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.

 

[haruspex]

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.

 

[Ray Vickson]

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.

 

[Interior]

Haruspex, yes I know that, but I simply don't know how to do that.
Can you give some clues?

 

[haruspex]

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?

 

[Ray Vickson]

Haruspex, yes I know that, but I simply don't know how to do that.
Can you give some clues?

Look at post #3.