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
The difference between g(x) and f(x) must be between 1 and 3, for all x in the interval. What's more, g is continuous, so it can't be greater than f(x) for some values of x, and less than f(x) for other values of x. So either for all x in [0,1], or for all x in [0,1].