1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show that the map is continuous

  1. May 2, 2017 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations


    3. The attempt at a solution

    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.

     
  2. jcsd
  3. May 2, 2017 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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.
     
  4. May 2, 2017 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
    Last edited: May 3, 2017
  5. May 3, 2017 #4
    Haruspex, yes I know that, but I simply don't know how to do that.
    Can you give some clues?
     
  6. May 3, 2017 #5

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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?
     
  7. May 3, 2017 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Look at post #3.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Show that the map is continuous
  1. Continuous map (Replies: 25)

Loading...