Let C be the space of continuous functions from [a,b] to R (real numbers) with the metric:
d(f,g) = sup lf-gl
Is the interior of C a non-empty set?
I say it is an empty set, but I'm not sure about some of my steps. Let me know what you all think:
Outline of Proof
Suppose the interior is not empty and contains f.
Then, for some e>0, the open ball around f of radius e is contained in C.
But, because f is continuous, then if we break [a,b] up into small enough intervals such that on each interval [Xi, X(i+1)], there is a number Ci whereby:
If x is in [Xi, X(i+1)) -->lf(x) - Cil < e
Define a function g as:
g(x) = Ci if x is in [Xi, X(i+1))
then d(f,g) < e
so g is in the ball of radius e around f
so g is in C, but g is not continuous...contradiction.