show that map is continuous

Apr 2017
5
0
Denmark
Hello,

I wish to show that the map F: R^3 -> R^2 given by F(x,y,z)=(0,5*((e^x)+y),0,5*((e^x)-y)) is continuous.

I can argue that F is continuous because it consists of functions G(x,y)=0,5*((e^x)+y) and H(x,y)=0,5*((e^x)-y) which themselves are continuous.

But how can I show that the map is continuous by using the definition of continuity which involves the preimage?

(sorry but the latex-function is not working)

Appreciate the help.
 

Plato

MHF Helper
Aug 2006
22,506
8,663
I wish to show that the map F: R^3 -> R^2 given by F(x,y,z)=(0,5*((e^x)+y),0,5*((e^x)-y)) is continuous.
But how can I show that the map is continuous by using the definition of continuity which involves the preimage?
(sorry but the latex-function is not working)
LaTex is working: $F: R^3 \to R^2~\&~F(x,y,z)=(0.5*(e^x+y),~0.5*(e^x-y))$ (please do not use commas for decimal points)

For each $\varepsilon >0$ you need to produce a $\delta>0$ such that in $P\in\{(x,y,z) : (x-a)^2+(y-b)^2+(z-c)^2<\delta^2\}$
then $\|F(P)-F(a,b,c)\|<\varepsilon$. In other words. Given any circle centered at some point in $F(\mathbb{R}^3)$ then there is a sphere centered at the pre-image that maps entirely into the circle.
 
Apr 2017
5
0
Denmark
Hello and thanks for the response,

I am not quite sure how to do that (how do I produce such a delta?) Could you guide me in the right direction?

Thanks.
 
Last edited:
Apr 2017
5
0
Denmark
Hello again,

I don't know if I'm allowed to ask again But I really wish to understand how to solve this problem.
If someone could break it (thanks to Plato) down for me, I would certainly appreciate it. I am trying to understand the machinery behind this type of proof.

Thanks.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
There are a number of different, but equivalent, definitions of "continuous". What definition of "continuous" are you using?
 
Apr 2017
5
0
Denmark
I am using (or wish to use) the following definition:

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

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
Work backwards. Say $\left(\dfrac{e^x+y}{2},\dfrac{e^x-y}{2}\right) = (a,b)\in \mathbb{R}^2$. Then $x=\ln |a+b|$ and $y = a-b$.
 
Apr 2017
5
0
Denmark
Hmmm ... ok so the "points" in \(\displaystyle f^{-1}(o)\) are the points described by \(\displaystyle x=ln(a+b)\), \(\displaystyle y=a-b\) and a \(\displaystyle z\).

Then I can try to find and open ball (and show the set is open)?
Is the z-coordinate specified?
 

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
The preimage would be $f^{-1}(O) = \left\{(\ln (a+b), a-b, z): z\in \mathbb{R}, (a,b)\in O, a+b>0 \right\}$. The preimage for a single point would be: $f^{-1}(a,b) = \left\{(\ln(a+b),a-b,z): z\in \mathbb{R}, a+b>0\right\}$. Note that, for example, $f^{-1}(0,0) = \emptyset$. All points with nonempty preimage have preimages that are lines.