Given a metric space (X, dx), let C(X) denote the set of continuous functions from X to . For two functions in C(X), define
d( ) = sup | | for x in X.
Prove that C(X) is path-connected.
Thanks!
We need to show two things.
First we need a path between and two points.
Remember a path is
note that and .
Now we need to show that the path stays in C(X)
If you know the right theorems i.e products and sums of continous functions are continous then you are done, but my guess since the gave you the metric is that you need to show is continous.
Either use a seqence or delta epsilon proof. for cont.
I hope this helps.