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metric spaces and continuity

Does any one know how to prove c [0,1] x c [0,1]-> c [0,1] defined by (phi (f, g))(x) =f (x).g (x) is continous using the epsilon delta definition in matric spaces I'm so lost
Reply 1
Avatar for james22
james22
16
Do you have to use the epsilon-delta defiition, or can you use the open sets definition?
Reply 2
Avatar for omassey
omassey
OP
I think we can use theopen sets one too
Reply 3
Avatar for james22
james22
16
Original post by omassey
I think we can use theopen sets one too


What is the full question? There is some stuff you aren't telling us. What are f and g?
Original post by james22
What is the full question? There is some stuff you aren't telling us. What are f and g?
I suspect that c [0,1] is supposed to be the set of cts functions on [0,1]. So (f, g) would be a typical element of c[0,1] x c[0,1] (i.e. f, g are both cts functions on [0,1]).

Thing is,we kind of need to know what metric we should be using to measure the distance between two functions (all the likely candidates are equivalent, so in a way it doesn't really matter, but still)...
Reply 5
Avatar for james22
james22
16
Original post by DFranklin
I suspect that c [0,1] is supposed to be the set of cts functions on [0,1]. So (f, g) would be a typical element of c[0,1] x c[0,1] (i.e. f, g are both cts functions on [0,1]).

Thing is,we kind of need to know what metric we should be using to measure the distance between two functions (all the likely candidates are equivalent, so in a way it doesn't really matter, but still)...


Didn't notice the c's, that makes the question very different and your right about the metric. There isn't really a standard one with functions like there is on the reals.
Reply 6
Avatar for omassey
omassey
OP
Oh sorry the metric is D2 and yes f and g are functions

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