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## 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

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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.