What is the region of functions in C[0,1] defined by the sup metric?

[rolylane]

Hi there

I have a proof that I need to try to work out but I'm not really getting too far and need help if you could at all. 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 B={g Є C[0,1]: 1 ≤ d(g,f) ≤ 3}
Describe the region in which the functions in B have their graphs
Let h:[0,1]→R be the function given by h(x)= 6x and Let k:[0,1]→R be the function given by k(x)=2+x/2. Is h Є B? Is k Є B?

Any help at all would be so great
Cheers

 

[andytoh]

B = region between the parabolas y=x^2+5 and y=x^2+3 and region between the parobolas y=x^2+1 and y=x^2-1. Draw a picture.

 

[rolylane]

B = region between the parabolas y=x^2+5 and y=x^2+3 and region between the parobolas y=x^2+1 and y=x^2-1. Draw a picture.

I'll try that and see how I get on then. Thanks so much for your suggestion.

Cheers!