Proving Continuity of F:XxI->I with Continuous Functions

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How can I show that F:X\times I\to I given by F(x,t)=(1-t)f(x)+tg(x) is continuous, given that f:X\to I and g:X\to I 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!)
 
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what do you know about continuous functions? compositions of them are still continuous, as are sums, products,...
 
mathwonk said:
what do you know about continuous functions? compositions of them are still continuous, as are sums, products,...
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.
 
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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