Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

X is any topological space

  1. Mar 7, 2005 #1
    How can I show that [tex]F:X\times I\to I[/tex] given by [tex]F(x,t)=(1-t)f(x)+tg(x)[/tex] is continuous, given that [tex]f:X\to I[/tex] and [tex]g:X\to I[/tex] are continuous (here I is the unit interval [0,1]). It seems that F is continuous, but I want to show that explicitly. Any help appreciated! X is any topological space.
    (I wasn't sure what section to put this in - sorry!)
     
    Last edited: Mar 7, 2005
  2. jcsd
  3. Mar 8, 2005 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    The inverse image of an open set is open...
     
  4. Mar 8, 2005 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    what do you know about continuous functions? compositions of them are still continuous, as are sums, products,....
     
  5. Mar 8, 2005 #4
    I'm willing to take that compositions of continuous functions are continuous without proof. I know that sums and products are continuous, but only when the domain is some subset of R^n. Does this carry over to any domain? If that's the case, then there is nothing to show.
    I guess my main hangup is determining the inverse image of say (a,b). I'm going to try a specific example and see if that helps.
     
  6. Mar 8, 2005 #5

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    think about how you prove sums and products are continuous.

    i.e. the addition and multiplication mapp from RxR to R are cotinuous. then if you have two continuous maps f:X-->R and g:X-->R you get one continuous map

    (f,g):X-->RxR, and you comkpose with addition or multiplication. so it has nothing to do with the domain of the functions.

    i agree there is nothing to do for your problem. but it can only be good to look at examples.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?