Proving Continuity of g: C0([0, 1]) x [0, 1]-->R

[metder]

Homework Statement

Let C⁰([0, 1]) be the set of continuous functions on the interval [0, 1] with the supremum topology. Prove that the map given by g: C⁰([0, 1]) x [0, 1]-->R given by g(f, a) = f(a) is continuous.

The Attempt at a Solution

I was originally thinking that maybe I could use the Urysohn lemma to show continuity, but I could not figure out how to make that work in a proof. The simpler method of looking at pre-images of g has also not yielded any insight so far. Any help would be appreciated.

 

[LCKurtz]

Can't you just show it directly? Given f and a you want to make |f(a) - g(b)| small if g is close to f in the sup norm and a is close to b. Try adding and subtracting f(b).

 

[Dick]

Ok, you want to show |f1(a1)-f(a)| is close to 0 where f1 is close to f in the supremum topology and a1 is close to a. Did you use that f is uniformly continuous on [0,1] since [0,1] is compact?

 

[metder]

Yeah, you're right. I was over thinking the problem. Thanks for the help.