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Thread: show that map is continuous

  1. #1
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    show that map is continuous

    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.
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  2. #2
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    Re: show that map is continuous

    Quote Originally Posted by Interior View Post
    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.
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  3. #3
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    Re: show that map is continuous

    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 by Interior; May 1st 2017 at 12:31 PM.
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  4. #4
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    Re: show that map is continuous

    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.
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  5. #5
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    Re: show that map is continuous

    There are a number of different, but equivalent, definitions of "continuous". What definition of "continuous" are you using?
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  6. #6
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    Re: show that map is continuous

    I am using (or wish to use) the following definition:

    "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."
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  7. #7
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    Re: show that map is continuous

    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 $y = a-b$.
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  8. #8
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    Re: show that map is continuous

    Hmmm ... ok so the "points" in f^{-1}(o) are the points described by x=ln(a+b), y=a-b and a z.

    Then I can try to find and open ball (and show the set is open)?
    Is the z-coordinate specified?
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  9. #9
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    Re: show that map is continuous

    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.
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