# show that map is continuous

#### Interior

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
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.

#### Interior

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:

#### Interior

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
There are a number of different, but equivalent, definitions of "continuous". What definition of "continuous" are you using?

#### Interior

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
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 $y = a-b$.

#### Interior

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
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.