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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
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  3. Mar 8, 2005 #2

    matt grime

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

    mathwonk

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    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

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    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.
     
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