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

[Euclid]

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!)

 

[matt grime]

The inverse image of an open set is open...

 

[mathwonk]

what do you know about continuous functions? compositions of them are still continuous, as are sums, products,...

 

[Euclid]

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.

 

[mathwonk]

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.