Proving Open/Closed Sets: Functional Analysis in C [0,1]

[patricia-donn]

Hello
Would anyone out there be able to help me with a problem I'm having? I have to prove that a function is open and that another is closed. The question is:

Consider C [0,1] with the sup metric. Let f:[0,1]→R be the function given by f(x)=x²+2
Let A={g Є C[0,1]: d(g,f) > 3}. Prove that A is an open set
Let B={g Є C[0,1]: 1 ≤ d(g,f) ≤ 3}. Prove that B is a closed set

I'm new to all of this and just don't know what to do even with the f(x)=x²+2 part so if anyone out there can shed some light, I'd be really grateful!

Thanks

 

[morphism]

You're trying to prove that sets are open and closed, not functions. I suggest you sketch the graph of f. Here, d(f,g) represents the maximum distance between two continuous functions f and g. So play around with the sketch, and then try to prove your observations.

 

[patricia-donn]

You're trying to prove that sets are open and closed, not functions. I suggest you sketch the graph of f. Here, d(f,g) represents the maximum distance between two continuous functions f and g. So play around with the sketch, and then try to prove your observations.

Thank you for your advice, I'll try that and see how I get on

Thanks again